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THE  LOWELL  LECTURES 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


THE  MACMILLAX  COMPANY 

NEW  YORK   •   BOSTON  ■   CHICAGO  •  DALLAS 
ATLANTA  •   SAN  FRANCISCO 

MACMILLAN  &  CO.,  Limited 

LONDON  •   BOMBAY  •  CALCUTTA 
MELBOURNE 


THE  MACMILLAN  CO.  OF  CANADA,  Ltd 

TORONTO 


PHOTOGRAPHS  OF  SOUND  WAVES 


THE  SCIENCE 
OF  MUSICAL  SOUNDS 


6F  PRi 


BY 


DAYTON  CLARENCE  MILLER,  D.Sc. 

PROFESSOR  OF  PHYSICS 
CASE  SCHOOL  OF  APPLIED  SCIENCE 


Neto  gork 

THE  MACMILLAN  COMPANY 
1916 

All  rights  reserved 


Copyright,  191G, 
By  the  MACMILLAN  COMPANY. 

Set  up  and  printed.    Published  March,  1916. 


J.  8.  Cashing  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE 


A  SERIES  of  eight  lectures  was  given  at  the  Lowell  Insti- 
tute in  January  and  February,  1914,  under  the  general  title 
of  "Sound  Analysis."  These  lectures  have  been  rewritten 
for  presentation  in  book  form,  and  "  The  Science  of  Musical 
Sounds  "  has  been  chosen  for  the  title  as  giving  a  better  idea  of 
the  contents.  They  appear  substantially  as  delivered,  though 
some  slight  additions  have  been  made  and  much  explanatory 
detail  regarding  the  experiments  and  illustrations  has  been 
omitted.  The  important  additions  relate  to  the  tuning  fork 
in  Lecture  II  and  harmonic  analysis  in  Lecture  IV,  while  two 
quotations  are  added  in  the  concluding  section  of  Lecture  VIII. 

A  course  of  scientific  lectures  designed  for  the  general  public 
must  necessarily  consist  in  large  part  of  elementary  and  well 
known  material,  selected  and  arranged  to  develop  the  principal 
line  of  thought.  It  is  expected  that  lectures  under  the  aus- 
pices of  the  Lowell  Institute,  however  elementary  their  foun- 
dation, will  present  the  most  recent  progress  of  the  science. 
The  explanations  of  general  principles  and  the  accounts  of 
recent  researches  must  be  brief  and  often  incomplete ;  never- 
theless it  is  hoped  that  the  lectures  in  book  form  will  furnish 
a  useful  basis  for  more  extended  study,  and  to  further  this  end 
they  are  supplemented  by  references  to  sources  of  additional 
information.  The  references  are  collected  in  an  appendix, 
citations  being  made  by  numbers  in  the  text  corresponding  to 
the  numbers  in  the  appendix. 

It  is  further  expected  that  such  lectures  will  be  accompanied 
by  experiments  and  illustrations  to  the  greatest  possible  de- 
gree ;   the  nature  and  extent  of  this  illustrative  material  is 

V 


PREFACE 


shown  as  well  as  may  be  by  the  aid  of  diagrams  and  pictures, 
nearly  all  of  which  have  been  especially  prepared,  and  much 
care  has  been  taken  to  make  them  as  expressive  as  possible  of 
the  original  demonstrations  and  explanations. 

The  methods  and  instruments  used  in  sound  analysis  by  the 
author,  and  many  of  the  results  of  such  work,  were  described 
in  the  lectures  in  advance  of  other  publication  and  it  is  the 
intention  to  supplement  the  brief  accounts  here  given  by  more 
detailed  reports  in  scientific  journals. 

The  author  is  greatly  indebted  to  many  friends  for  the 
kindly  interest  shown  during  the  progress  of  the  experimental 
work  here  described  ;  and  he  is  especially  under  obligation  to 
Professor  Frank  P.  Whitman  of  Western  Reserve  University, 
and  to  Mr.  Eckstein  Case  and  Professor  John  M.  Telleen  of 
Case  School  of  Applied  Science,  for  many  helpful  suggestions 
received  while  the  manuscript  was  in  preparation. 

DAYTON  C.  MILLER. 

Cleveland,  Ohio, 
July,  1915. 


vi 


CONTENTS 


LECTURE  I 

SOUXI)  WAVES,  SIMPLE  HARMONIC  MOTION, 
XOISE  AND  TONE 

PAGE 

Introduction  —  Sound  defined  —  Simple  harmonic  motion  and  curve 

—  Wave  motion  —  The  ear  —  Noise  and  tone         ....  1 


LECTURE  II 

CHARACTERISTICS  OF  TONES 

Pitch  —  The  tuning  fork  —  Determination  of  pitch  by  the  method  of 
beats  —  Optical  comparison  of  pitches  —  The  clock-fork  —  Pitch 
limits  —  Standard  pitches  —  Intensity  and  loudness  —  Acoustic 
properties  of  auditoriums  —  Tone  quality — Law  of  tone  quality 
—  Analysis  by  the  ear  26 


LECTURE  III 

METHODS  OF  RECORDING  AND  PHOTOGRAPHING 
SOUND  WAVES 

The  diaphragm  —  The  i»honautograph  —  The  manometric  flame  — 
The  oscillograph  —  The  phonograph  —  The  phonodeik  —  The 
demonstration  phonodeik  —  Determination  of  pitch  with  the  pho- 
nodeik—  Photographs  of  compression  waves  70 


LECTURE  IV 

ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 

Harmonic  analysis  —  Mechanical  harmonic  analysis  —  Amplitude  and 
phase  calculator — Axis  of  a  curve  —  Enlarging  the  curves  —  Syn- 
thesis of  harmonic  curves  —  The  complete  process  of  harmonic 
analysis  —  Example  of  harmonic  analysis  —  Various  types  of 
harmonic  analyzers  and  synthesizers  —  Arithmetical  and  graph- 
ical methods  of  harmonic  analysis  —  Analysis  by  inspection  — 

Periodic  and  non-periodic  curves  92 

vii 


CONTENTS 


LECTURE  V 

INFLUENCE  OF  HORN  AND  DIAPHRAGM  ON  SOUND 
WAVES,  CORRECTING  AND  INTERPRETING  SOUND 
ANALYSES 

PAGE 

Errors  in  sound  records  —  Ideal  response  to  sound  —  Actual  response 
to  sound  —  Response  of  the  diaphragm  —  Chladni's  sand  figures 

—  Free  periods  of  the  diaphragm  —  Influence  of  the  mounting  of 
the  diaphragm  —  Influence  of  the  vibrator  —  Influence  of  the  horn 

—  Correcting  analyses  of  sound  waves —  Graphical  presentation  of 
sound  analyses  —  Verification  of  the  method  of  correction  — 
Quantitative  analysis  of  tone  quality  142 


LECTURE  VI 
TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 

Generators  and  resonators —  Resonance  —  Effects  of  material  on  sound 
waves  —  Beat-tones  —  Identification  of  instrumental  tones — The 
tuning  fork  —  The  flute  —  The  violin  —  The  clarinet  and  the  oboe 
—  The  horn  —  The  voice  —  The  piano —  Sextette  and  orchestra  — 
The  ideal  musical  tone  —  Demonstration  175 


LECTURE  VII 
PHYSICAL  CHARACTERISTICS  OF  THE  VOWELS 

The  vowels  —  Standard  vowel  tones  and  words  —  Photographing, 
analyzing,  and  plotting  vowel  curves  —  Vowels  of  various  voices 
and  pitches  —  Definitive  investigation  of  one  voice  —  Classifica- 
tion of  vowels — Translation  of  vowels  with  the  phonograph  — 
Whispered  vowels  —  Theory  of  vowel  quality        ....  215 


LECTURE  VIII 

SYNTHETIC  VOWELS  AND  WORDS,  RELATIONS  OF 
THE  ART  AND  SCIENCE  OF  MUSIC 

i 

Artificial  and  synthetic  vowels  —  Word  formation  —  Vocal  and  instru- 
mental tones — "Opera  in  English"  —  Relations  of  the  art  and 
science  of  music  244 

Appendix  —  References  271 

Index  ,       .  281 


viii 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


LECTURE  1 

SOUND  WAVES,  SIMPLE  HARMONIC  MOTION, 
NOISE  AND  TONE 

Introduction 

We  are  beings  with  several  senses  through  which  we  come 
into  direct  relation  with  the  world  outside  of  ourselves. 
Through  two  of  these,  sight  and  hearing,  we  are  able  to 
receive  impressions  from  a  distance  and  through  these  only 
do  the  fine  arts  appeal  to  us ;  through  sight  we  receive  the 
arts  of  painting,  sculpture,  and  architecture,  and  through 
hearing,  the  arts  of  poetry  and  music. 

Undoubtedly  music  gives  greater  pleasure  to  more  people 
than  does  any  other  art,  and  probably  this  enjoyment  is  of 
a  more  subtle  and  pervading  nature ;  every  one  enjoys 
music  in  some  degree,  and  many  enjoy  it  supremely.  Sound 
is  also  of  the  greatest  practical  importance ;  we  rely  upon  it 
continually  for  the  protection  of  our  lives,  and  through  talk- 
ing, which  is  but  making  sounds  according  to  formula,  we 
receive  information  and  entertainment.  These  facts  give 
ample  justification  for  studying  the  nature  of  sound,  the 
material  out  of  which  music  and  speech  are  made. 

The  study  of  sounds  in  language  is  as  old  as  the  human 
race,  and  the  art  of  music  is  older  than  tradition,  but  the 
science  of  music  is  quite  as  modern  as  the  other  so-called 
modern  sciences.    Sound  being  comparatively  a  tangible 

B  1 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


phenomenon,  and  so  intimately  associated  with  the  very 
existence  of  every  human  being,  one  would  expect  that  if 
there  are  any  unknown  facts  relating-  to  it,  a  large  number  of 
investigators  would  be  at  work  trying  to  discover  them. 
There  has  been  in  the  past,  as  there  is  now,  a  small  number  of 
enthusiastic  workers  in  the  field  of  acoustics  who  have  ac- 
complished much ;  but  it  is  no  doubt  true  that  this  science 
has  received  less  attention  than  it  deserves,  and  especially 
may  this  be  said  of  the  relation  of  acoustics  to  music. 

Sound  Defined 

Sound  may  be  defined  as  the  sensation  resulting  from  the 
action  of  an  external  stimulus  on  the  sensitive  nerve  ap- 
paratus of  the  ear ;  it  is  a  species  of  reaction  to  this  external 
stimulus,  excitable  only  through  the  ear,  and  distinct  from 
any  other  sensation.  Atmospheric  vibration  is  the  normal 
and  usual  means  of  excitement  for  the  ear ;  this  vibration 
originates  in  a  source  called  the  sounding  body,  which  is 
itself  always  in  vibration. 

The  source  may  be  constructed  especially  to  produce 
sound ;  in  a  stringed  instrument  the  string  is  plucked  or 
bowed  and  its  vibration  is  transferred  to  the  soundboard, 
and  this  in  turn  impresses  the  motion  upon  a  larger  mass  of 
air ;  in  the  flute  and  other  wind  instruments  the  air  is  set 
in  motion  directly  by  the  breath.  The  vibration  often  origi- 
nates in  bodies  not  designed  for  producing  sounds,  as  is 
illustrated  by  the  squeak  and  rumble  of  machinery. 

The  physicist  uses  the  word  sound  to  designate  the  vibra- 
tions of  the  sounding  body  itself,  or  those  which  are  set  up  by 
the  sounding  body  in  the  air  or  other  medium  and  which 
are  capable  of  directly  affecting  the  ear  even  though  there  is 
no  ear  to  hear. 

2 


/ 


THE  NATURE  OF  SOUND 


There  are  numerous  experiments  which  demonstrate  that 
a  sounding  body  vibrates  vigorously.  When  a  tuning  fork, 
Fig.  1,  is  struck  with  a  soft  felt  hammer,  it  gives  forth  a 
continuous  sound.  The  fork  vibrates  transversely  several 
hundred  times  a  second,  though  the  distance  through  which 
it  moves  is  only  a  few  thousandths  of  an  iii-^h.  These  move- 
ments, which  are 
too  minute  and 
rapid  to  be  ap- 
preciated by  the 
eye,  may  be  made 
evident  by  means 
of  a  pith-ball  pen- 
dulum adjusted 
to  rest  lightly 
against  the  prong  ; 
when  the  fork  is 
sounding,  the  ball 
is  violently- 
thrown  aside. 

The  powerful 
longitudinal  vi- 
brations of  a 
metal  rod  may  be 
exhibited  in  a  like 

manner  with  apparatus  arranged  as  shown  in  Fig.  2. 
If  a  piece  of  rosined  leather  is  drawn  along  the  rod  a 
loud  tone  is  emitted,  and  the  ivory  ball  resting  lightly 
against  the  end  is  thrown  high  in  the  air.  At  the  center 
of  the  rod  the  molecules  are  at  rest,  forming  a  node, 
while  at  the  ends  the  vibrations  are  of  relatively  large 
amplitude.    The  molecules  vibrate  hundreds  of  times  per 

3 


Tuning  fork  and  pith  ball  for  demonstration 
of  vibration. 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


second,  and  the  comparatively  feeble  movements  of  single 
molecules,  by  their  cumulative  effects  in  the  mass,  develop 
enormous  forces,  equivalent  to  a  tensile  force  of  several 
tons.  If  a  glass  tube,  held  at  its  middle,  is  rubbed  with  a 
piece  of  wet  cloth,  the  longitudinal  vibrations  are  often  so 
vigorous  as  to  cause  the  tube  to  separate  into  many  pieces. 
A  glass  bell  may  be  set  into  transverse  vibration  by  bow- 


FiG.  2.    Longitudinally  vibrating  rod. 


ing  across  the  edge.  The  vibrations  cause  periodic  deforma- 
tions of  the  shape,  from  a  circle  to  an  ellipse  and  back  to 
the  circle.  The  circumference  must  vibrate  in  at  least 
four  segments,  with  the  formation  of  loops  and  nodes. 
Balls,  suspended  from  a  revolving  support,  as  shown  in 
Fig.  3,  may  rest  against  the  surface  of  the  bell  and  may  be 
used  to  locate  the  loops  and  nodes.  The  vibration  may 
become  so  violent  as  to  shatter  the  glass. 

4 


THE  NATURE  OF  SOUND 


The  vibrations  of  the  source  produce  various  physical 
effects  in  the  surrounding  air,  such  as  displacements,  ve- 
locities, and  accelerations,  and  changes  of  density,  pressure, 
and  temperature ;  because  of  the  elasticity  of  the  air,  these 
displacements  and  other  phenomena  occur  periodically  and 
are  transmitted  from  particle  to  particle  in  such  a  manner 


Fig.  3.    Glass  bell. 


that  the  effects  are  propagated  outward  from  the  source 
in  radial  directions.  These  disturbances  of  all  kinds,  as 
they  exist  in  the  air  around  a  sounding  body,  constitute 
sound  waves.  The  velocity  of  sound  is  about  1132  feet  per 
second  when  the  temperature  of  the  air  is  70°  F.,  a  tempera- 
ture common  in  auditoriums  ;  at  the  freezing  temperature, 
32°  F.,  it  is  about  1090  feet  per  second.  Musical  sounds  of 
different  pitches  are  all  propagated  in  the  open  air  with  the 

5 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


same  velocity.  Explosive  sounds  and  sounds  confined, 
as  in  tubes,  are  propagated  with  different  velocities.^  Wave 
disturbances  may  be  transmitted  by  solid  and  liquid  as 
well  as  gaseous  matter,  but  our  present  study  relates  mainly 
to  what  may  be  heard,  and  the  explanations  are  limited  for 
the  most  part  to  certain  features  of  waves  in  air,  and  par- 
ticularly to  the  nature  of  the  movements  of  the  air  parti- 
cles when  transmitting  musical  sounds. 

Simple  Harmonic  Motion  and  Curve 

The  simplest  possible  type  of  vibration  which  a  particle 
of  elastic  matter  of  any  kind  may  have  is  called  simple 
harmonic  motion;  it  takes  place  in  a  straight  line,  the  mid- 
dle of  which  is  the  position  of  rest  of  the  particle ;  when 
the  particle  is  displaced  from  this  position,  elasticity 
develops  a  force  tending  to  restore  it,  which  force  is  directly 
proportional  to  the  amount  of  the  displacement;  if  the  displaced 
particle  is  now  freely  released,  it  will  vibrate  to  and  fro 
with  simple  harmonic  motion.  The  name  originated  in 
the  fact  that  musical  sounds  in  general  are  produced  by 
complex  vibrations  which  can  be  resolved  into  component 
motions  of  this  type. 

Other  forces  than  those  of  elasticity  may  act  in  the  manner 
described,  as  for  instance  the  action  of  the  force  of  gravity 
on  the  bob  of  a  pendulum ;  if  the  bob  is  considered  as  swing- 
ing in  a  straight  line,  it  has  simple  harmonic  motion,  which 
is  also  called  pendular  motion. 

Simple  harmonic  motion  has  several  evident  features : 
it  takes  place  in  a  straight  line ;  it  is  vibratory,  moving  to 
and  fro ;  it  is  periodic,  repeating  its  movements  regularly ; 
there  are  instants  of  rest  at  the  two  extremes  of  the  move- 
ment ;   starting  from  rest  at  one  extreme  the  movement 

6 


SIMPLE  HARMONIC  MOTION 


quickens  till  it  reaches  its  central  point,  after  which  it 
slackens  in  reverse  order,  till  it  comes  to  rest  at  the  other 
extreme.  The  speed  of  the  particle  so  moving,  the  rate 
at  which  the  speed  changes,  and  other  features  are  very- 
important  in  a  complete  study  of  simple  harmonic  motion, 
but  for  our  purpose  we  need  give  only  a  few  simple  defini- 
tions. 

The  frequency  of  a  simple 
harmonic  motion  is  the  num- 
ber of  complete  vibrations  to 
and  fro  per  second  ;  the  period 
is  the  time  required  for  one 
complete  vibration  ;  the  am- 
plitude is  the  range  on  one 
side  or  the  other  from  the 
middle  point  of  the  motion, 
therefore  it  is  half  the  ex- 
treme range  of  vibration  ; 
the  phase  at  any  instant  is 
the  fraction  of  a  period  which 
has  elapsed  since  the  point 
last  passed  through  its  mid- 
dle position  in  the  direction 
chosen  as  positive. 

Simple  harmonic  motion  is  approximated  in  various  me- 
chanical movements,  while  a  few  simple  machines  reproduce 
it  exactly ;  -  this  reproduction  is  always  accomplished  b}^  a 
transformation  of  uniform  motion  in  a  circle  into  rectilinear 
motion.  The  pin-and-slot  device  has  a  slotted  frame  s, 
Fig.  4,  which  is  movable  up  and  down  only;  the  pin  p 
of  the  crank  c  moves  in  the  slot ;  when  the  crank  is  turned 
with  uniform  angular  speed,  the  frame  and  all  rigidly  at- 

7 


Fig.  4.    Simple  harmonic  motion  from 
mechanical  movement. 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


tached  parts,  such  as  the  point  P,  move  with  simple  har- 
monic motion.  The  usual  starting  point  for  this  motion  is 
the  middle  position  of  P  when  it  is  about  to  move  upward, 
that  is,  when  the  crank  is  horizontal  with  the  pin  at  the 
extreme  right  and  about  to  turn  counterclockwise.  One 
complete  vibration  is  produced  when  the  crank  makes  one 

revolution  and  the  point 
P  moves  from  its  mid- 
position  to  the  extreme 
upper  position,  down  to 
the^  lower  extreme,  and 
back  to  mid-position. 
The  period  is  the  time 
required  for  the  complete 
vibration,  that  is,  for  one 
revolution  of  the  crank ; 
the  phase  at  any  instant 
is  the  fraction  of  a  period 
which  has  elapsed  since 
the  point  last  passed 
through  the  starting 
point,  and  is  often  ex- 
pressed by  the  number 
of  degrees  through  which 
the  crank  has  turned  in  the  interval,  as  is  further  illustrated 
in  Fig.  6 ;  the  amplitude  is  measured  by  half  the  extreme 
movement,  that  is,  by  the  length  of  the  crank  from  center 
to  pin.  This  device  is  used  in  several  of  the  harmonic 
synthesizers  described  in  Lecture  IV. 

In  treatises  on  mechanics  simple  harmonic  motion  is  often 
defined  as  the  projection  of  uniform  motion  in  a  circle  upon 
a  diameter  of  the  circle ;  this  definition  is  illustrated  by  the 

8 


1 

■  I 

n 



Fig.  5. 


Relation  of  simple  harmonic  and 
circular  motion. 


SIMPLE  HARMONIC  MOTION 


form  of  the  pin-and-slot  apparatus  shown  in  Fig.  5.  Turn- 
ing the  crank  on  the  back  of  the  apparatus  causes  the  point  P 
in  the  diameter  to  move  up  and  down  with  a  true  harmonic 


0  Phase  0      HPkase        90      ^Phase    225  J^Phase 

Fig.  6.    Phases  of  simple  harmonic  motion. 


315 


motion  when  the  point  p  in  the  circle  revolves  with  uniform 
speed;  the  two  points  are  always  in  the  same  horizontal 
line,  or  the  point  in  the  diameter  is  always  the  projection 
of  the  one  in- the  circle;  Fig.  6  illustrates  the  motion  in 
various  phases. 

A  crank  pin  p  is  pivoted  in 
the  center  of  a  rod  AB,  Fig. 
7 ;  the  ends  of  the  rod  are 
pivoted  to  sliders  which 
move  in  two  perpendicular, 
straight  grooves ;  when  the 
crank  is  turned  with  uniform 
speed,  both  of  the  points  A 
and  B  move  with  simple  har- 
monic motion.  The  dis- 
placement of  either  slider 
from  its  central  position  is 
always  twice  the  displace- 
ment of  the  projection  of  the 
point  p  on  the  corresponding 

Fig.  7.    Simple  harmonic  motion  from 
groove.  mechanical  movement. 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


A  simple  harmonic  motion  can  be  obtained  without  the 
friction  of  sUders  in  grooves  by  employing  a  pantograph  to 
give  the  arithmetical  mean  of  two  equal  and  opposite  cir- 
cular motions,  as  suggested  by  Everett. ^  When  the  crank  c, 
Fig.  8,  is  turned,  the  point  P  moves  up  and  down  in  a 
straight  line,  so  that  it  is  always  in  the  horizontal  line  con- 


FlG.  8.    Simple  harmonic  motion  from       Fig.  \K    Simple  harmonic  UKjtion  from 
mechanical  movement.  mechanical  movement. 


necting  the  points  A  and  B,  and  therefore,  when  the  wheels 
rotate  with  uniform  speed,  it  has  simple  harmonic  motion. 

A  simple  harmonic  motion  is  given  to  any  point  P  on 
the  circumference  of  a  wheel.  Fig.  9,  when  the  wheel  rolls 
with  uniform  speed  on  the  inside  of  an  annulus  a,  the  radius 
of  which  is  equal  to  the  diameter  of  the  wheel.  The  point 
P  is  always  in  the  horizontal  line  passing  through  the  point 
of  contact  a  of  the  wheel  and  annulus. 

10 


SIMPLE  HARMONIC  MOTION 


The  movement  of  a  sliding  block  connected  to  a  crank  by 
a  pitman  rod,  as  the  crosshead  of  an  engine,  has  a  distorted 
simple  harmonic  motion,  the  errors  of  which  may  be  cor- 
rected by  suitable  mechanism ;  Fig.  10  shows  a  device  due 
to  Smedley,^  having  two  crossheads,  Ci  and  Ci,  on  opposite 
sides  of  the  crank  pin  ;  during  the  motion  these  are  oppositely 
displaced  from  the  true  harmonic  positions,  and  the  errors 
are  equalized  by  a  system  of  levers  acting  on  the  central 


Fig.  10.    Simple  harmonic  motion  from  compensated  crosshead  movement. 

sliding  block  P,  which  receives  simple  hannonic  motion 
w^hen  the  crank  revolves  uniformly. 

A  simple  harmonic  motion  combined  with  a  uniform  mo- 
tion of  translation  traces  a  simple  harmonic  curve;  this 
condition  is  illustrated  by  a  pendulum  swinging  from  a  fixed 
point,  Fig.  11,  and  leaving  a  trace  on  a  sheet  of  paper  mov- 
ing underneath.  The  simple  harmonic  curve,  Fig.  12,  is 
perfectly  simple,  regular,  and  symmetrical ;  in  mathematical 
study  it  is  frequently  referred  to  as  a  sine  curve;  a  curve  of 
the  same  form  but  differing  in  phase  by  a  quarter  period,  or 
90°,  is  a  cosine  curve.    As  explained  in  the  next  section,  such 

11 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


a  curve  is  an  instantaneous  representation  of  the  condition 
of  motion  in  a  simple  wave.    Various  terms  used  with 

regard  to  simple 
harmonic  motion  are 
also  applicable  to 
the  curve ;  the  am- 
plitude is  the  height 
of  a  crest  above  the 
axis,  Fig.  12 ;  the 
period  is  the  time 
required  to  trace  one 
wave  length  consist- 
ing of  a  crest  and 
trough  ;  the  f re- 
„         ^    .       •    ,  u       •  quency  is  the  num- 

FiG.  11.     1  racing  a  simple  harmonic  curve.  ^  ^ 

ber  of  periods,  or 
wave  lengths  traced,  per  second ;  the  phase  varies  along  the 
axis,  passing  through  a  complete  cycle  in  one  wave  length ; 


ti;'  axis 

K    -> 

#      \    fe,       \  , 

/x=o                    \  X=180  / 

/x=360°  \  \ 

^  S.H.M. 

 >     \tramlatz\on       /  / 

\           \  /  / 
^              V'  / 

^           \  / 

\         '  \  / 

'                   ^---'^  — ^ 

Fig.  12.    Sine  and  cosine  curves. 


the  velocity  is  the  rate  of  translation  and  is  equal  to  the  wave 
length  multiplied  by  the  number  of  waves  per  second. 
Sine  curves  may  differ  considerably  in  appearance,  de- 

12 


WAVE  MOTION 


pending  upon  the  relation  of  amplitude  and  wave  length 
(frequency),  though  all  must  have  the  same  general  prop- 
erties and  be  equally  regular  and  simple.  All  the  curves 
shown  in  Fig.  13  are  simple  harmonic,  or  sine  curves,  and 
differ  only  in  amplitude  A  and  frequency  n,  the  relative 
values  of  these  quantities  being  shown  in  the  figure. 


n  =  l  A=4 


n  =  iO  A=40 


Fig.  13.    Sine  curves  of  various  dimensions. 


Wave  Motion 

The  essential  characteristic  of  wave  motion  is  the  continu- 
ous passing  onward  from  point  to  point  in  an  elastic  medium 

13 


THE  SCIENCE  OF  MUSICAL  SOUNDS 

of  a  periodic  vibration  which  is  maintained  at  the  source. 
These  vibrations,  being  periodic,  produce  a  series  of  waves 
following  each  other  at  regular  intervals,  the  speed  of 
propagation  depending  upon  the  elastic  properties  of  the 
medium.  There  are  two  distinct  motions  involved :  the 
vibration  of  the  individual  particles  about  their  positions  of 
rest  and  the  progressive  outward  movement  of  the  wave 
form.     The  source  of  a  wave  motion  may  be  a  disturbance 


Fig.  14.    Machine  for  illustrating  transverse  waves. 


of  any  type,  but  for  sound  waves  it  consists  of  vibratory 
movements  which  are  either  simple  harmonic  or  compounds 
of  such. 

A  simple  transverse  wave  motion  is  represented  by  the  wave 
machine  shown  in  Fig.  14 ;  the  successive  pendulums  are 
given  similar  periodic  transverse  vibrations  in  successive 
times  by  a  slider  s,  which  is  moved  from  left  to  right  by  turn- 
ing the  handle  h.  The  slider  produces  a  wave  crest  which 
moves  along  the  row  of  balls  and  disappears,  being  followed 
periodically  by  other  crests ;  the  velocity  of  wave  propaga- 


WAVE  MOTION 


tion  is  the  velocity  with  which  tlie  sUder  is  moved ;  the 
wave  length  is  the  actual  distance  I  from  crest  to  crest  of 
the  wave ;  amplitude,  period,  and  frequency  are  illustrated 
in  the  vibrations  of  the  pendulums. 

One  of  the  bars  to  which  is  attached  one  string  of  each 
bifilar  suspension  of  a  pendulum  may  be  shifted  lengthwise 
and  away  from  the  other  bar  so  that  the  pendulums  can 
vibrate  only  in  a  longitudinal  direction  ;  by  moving  a  slider 


Fig.  15.    Machine  for  illustrating  longitudinal  waves. 


of  a  second  form  5,  Fig.  15,  simple  harmonic  motions,  exactly 
the  same  as  before  except  that  they  are  in  the  direction  of 
propagation,  are  given  to  the  series  of  pendulums  ;  this  pro- 
duces, instead  of  the  crests  and  troughs  of  the  former  wave, 
condensations  and  rarefactions  in  the  spacing  of  the  particles 
which  follow  each  other  periodically,  and  moving  forward 
^^ith  the  velocity  of  wave  propagation  illustrate  a  longitu- 
dinal wave  motion.  In  this  case  the  wave  length  is  the  dis- 
tance from  one  condensation  to  the  next,  and  the  various 

15 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


other  characteristics  are  substantially  the  same  as  for  the 
transverse  wave.  In  fact,  one  type  of  wave  can  be  trans- 
formed into  the  other  and  back  again  by  merely  shifting 
one  of  the  suspending  bars  while  the  pendulums  are  vibrat- 
ing ;  this  turns  the  direction  of  vibration  of  each  pendulum 
without  disturbing  the  character  of  the  motion. 

The  simple  harmonic  curve  may  be  considered  an  instan- 
taneous representation  of  a  transverse  wave ;  it  shows  by 
its  shape  the  nature  of  the  periodic  vibration  and  exhibits 
the  displacements  and  conditions  of  motion  of  a  continuous 
series  of  particles  transmitting  the  wave.    In  Fig.  16,  A 

A  


•    m    •  •    •  • 


C.    .  .  

F  IG.  16.    Transverse  and  longitudinal  displacements. 

represents  a  row  of  particles  at  rest ;  if  a  transverse  wave  is 
being  transmitted,  the  particles  at  some  instant  will  be  dis- 
placed as  shown  in  B,  forming  a  harmonic  curve.  If  the 
displacements  are  of  the  same  amounts  but  occur  in  a  longi- 
tudinal direction,  upward  displacements  in  B  corresponding 
to  forward  displacements  in  C,  and  vice  versa,  there  results  a 
longitudinal  wave  of  condensation  and  rarefaction,  or  of 
pressure  changes ;  this  is  the  type  of  sound  waves  in  air. 
Sound  waves  usually  pass  outward  from  the  source  in  the 
form  of  expanding  surfaces  of  disturbance,  and  the  nature 
of  the  pressure  changes,  as  applied  to  surfaces,  may  be  illus- 
trated by  the  spacing  of  the  lines  in  Bj  Fig.  17.    The  rela- 

16 


WA\  E  MOTION 


tions  of  condensation  and  rarefaction  of  the  longitudinal 
wave  to  crest  and  trough  of  the  transverse  wave  are  shown 
in  both  Figs.  16  and  17. 

The  amounts  of  displacement,  the  amplitudes,  periods, 
frequencies,  and  velocities  of  propagation  are  defined  in  exactly 
the  same  way  in  the  two  types  of  waves ;  only  the  directions 
of  displacement  differ.  It  can  be  shown  that  both  kinds  of 
wave  motion  are  adequately  and  correctly  represented  by  the 
harmonic  curve.  The  curve  B,  Fig.  16,  conveys  to  the  eye  a 
much  clearer  idea  of  the  displacements  than  does  C,  though 


A 


B 


Fig.  17.    Wave  of  compression. 


the  displacements  of  the  successive  particles  are  of  exactly  the 
same  amount  in  both  instances.  Nearly  all  of  the  waves  to 
be  studied  in  these  lectures  are  of  the  longitudinal  type,  but 
they  will  be  represented  by  curves  of  transverse  displacement. 

As  will  be  more  fully  developed  in  later  lectures,  several 
simple  harmonic  motions  of  various  amplitudes,  frequencies, 
and  phases,  moving  in  the  same  or  different  directions,  often 
coexist,  producing  wave  motions  which  are  represented  by 
curves  of  very  complex  shapes. 

Sound  waves  in  solids  may  be  either  transverse  or  longitu- 
dinal, but  the  properties  of  liquids  and  gases  are  such  that 
only  .longitudinal  displacements,  or  pressure  changes,  can  be 
c  17 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


transmitted  as  wave  motions. 
The  transmission  of  a  longitudi- 
nal wave  through  a  solid  or  Uquid 
body  is  illustrated  by  the  col- 
hsion  balls,  Fig.  18.  Wlien  the 
medium  is  very  compressible, 
such  as  a  gas,  the  method  of 
propagation  is  better  shown  by 
the  apparatus.  Fig.  19,  which 
consists  essentially  of  a  long, 
flexible  spring  suspended  so  as 
to  move  horizontally ;  if  a  push 
of  compression  is  given  to  one 
end  of  the  spring,  it  will  be 
transmitted  as  a  wave  in  a  man- 
ner to  be  easily  followed  by  the 

Fig.  18.    Collision  balls. 

eye. 

An  illustration  of  two  simple  harmonic  motions  at  right 
angles  is  given  by  the  compound  pendulum  apparatus  shown 
in  Fig.  20,  the  bob  of  which  is  a  weight  carrying  a  glass  vessel 


U 

i 

II' 

^  1^ 

Fig.  19.    Apparatus  for  illustrating  a  wave  of  compression. 
18 


WAVE  MOTION 


containing  sand ;  as  the  pendulum 
swings  the  sand  flows  from  a  small 
aperture  in  the  bottom  of  the  vessel 
and  leaves  a  trace  on  the  paper  un- 
derneath. The  pendulum  may 
swing  to  and  fro  and  from  side  to 
side ;  for  the  first  movement  its 
length  is  k,  but  on  account  of  the 
arrangement  of  the  two  suspending 
strings,  its  length  for  sidewise  move- 
ment is  U ;  hence  the  periods  of  the 
two  movements  are  different  and 
the  bob  swings  in  a  peculiar  curve 
compounded  of  two  simple  move- 


Fig.  20.    Compound  pendulum. 


19 


Fig.  21.    Torsional  wave. 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


ments ;  the  curve  shown  in  the  figure  results  from  periods 
of  the  exact  ratio  of  2  :  3 ;  other  ratios  give  characteristic 
figures ;  these  curves  are  known  as  Lissajous^s  figures}^ 
Compound  harmonic  motion  of  this  kind  is  made  use  of 
in  accurate  tuning,  as  described  in  Lecture  II. 

Various  other  types  of  motion  besides  simple  harmonic 
may  generate  waves  ;  a  torsional  wave,  consisting  of  angular 
harmonic  motion,  may  be  transmitted  by  a  loaded  wire,  as 
show^n  in  Fig.  2L 

The  Ear 

Sound  has  been  defined  as  the  sensation  received  through 
the  ear,  and  the  definition  has  been  extended  to  include  the 
external  cause  of  the  sensation.  All  that  the  ear  perceives 
in  the  complex  music  of  a  grand  opera  or  of  a  symphony 
orchestra  is  contained  in  the  wave  motion  of  the  air  consist- 
ing of  periodic  changes  in  pressure  and  completely  repre- 
sented by  motion  of  one  dimension,  that  is,  by  motion  con- 
fined to  a  straight  fine  ;  or,  as  Lord  Kelvin  has  expressed  it, 
sound  is    a  function  of  one  variable." 

That  motion  of  one  dimension  is  capable  of  producing 
these  sounds  is  amply  proved  by  the  talking  machine;  in 
the  cylinder  type  of  machine  the  tracing  point  moves  up 
and  down,  and  gives  a  backward-and-forward  motion  to  the 
diaphragm,  each  point  of  which  moves  in  a  straight  line ; 
the  resulting  wave  of  compression  is  transmitted  by  the  air 
to  the  eardrum.  The  telephone  is  another  demonstration 
of  the  same  fact.  Some  of  the  disk  types  of  talking  machines 
not  only  illustrate  the  movement  in  one  direction,  but  also 
demonstrate  that  a  transverse  vibration  on  the  record  is 
transformed,  through  the  needle  and  connecting  levers, 
into  an  equivalent  longitudinal  motion  at  the  diaphragm  of 
-the  sound  box.    It  is  marvelous  that  complex  musical  re- 

20 


NOISE  AND  TONE 

suits  can  be  produced  by  such  seemingly  simple  mechanical 
means. 

The  functioning  of  the  ear,  which  is  a  wonderfully  com- 
plex organ,  Fig.  22,  is  but  imperfectly  understood ;  physi- 
ologists are  studying  its  structure,  and  psychologists  are  in- 
vestigating the  manner  of  the  reception  and  perception  of 
the  sensation  of  sound  ;  the  study  of  these  most  interesting 
questions  is  quite  outside  of  the  province  of  these  lectures, 
which  is  confined  to  the  physics  of  that  which  may  be  heard. 


Fig.  22.    Model  of  the  ear,  dissected. 


that  is,  to  sound  waves  as  they  exist  in  the  air  and  to  their 
sources. 

The  ear  divides  sounds  roughly  into  two  classes  :  noises, 
which  are  disagreeable  or  irritating,  and  tones,  which  are 
received  with  pleasure  or  indifference. 

Noise  and  Tone 
Noise  and  tone  are  merely  terms  of  contrast,  in  extreme 
cases  clearly  distinct,  but  in  other  instances  blending ; 
the  difference  between  noise  and  tone  is  one  of  degree.  A 
simple  tone  is  absolutely  simple  mechanically;  a  musical 
tone  is  more  or  less  complex,  but  the  relations  of  the  com- 

21 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


ponent  tones,  and  of  one  musical  sound  to  another,  are 
appreciated  by  the  ear ;  noise  is  a  sound  of  too  short  dura- 
tion or  too  complex  in  structure  to  be  analyzed  or  understood 
by  the  ear. 

The  distinction  sometimes  made,  that  noise  is  due  to  a 
non-periodic  vibration  while  tone  is  periodic,  is  not  sufficient ; 
analysis  clearly  shows  that  many  so-called  musical  tones 
are  non-periodic  in  the  sense  of  the  definition,  and  it  is 
equally  certain  that  noises  are  as  periodic  as  are  some  tones. 
In  some  instances  noises  are  due  to  a  changing  period,  pro- 
ducing the  efTect  of  non-periodicity ;  but  by  far  the  greater 
number  of  noises  which  are  continuous  are  merely  complex 
and  only  apparently  irregular,  their  analysis  being  more  or 
less  difficult. 

The  ear,  because  of  lack  of  training  or  from  the  absence 
of  suitable  standards  for  comparison  or  perhaps  on  account 
of  fatigue,  often  fails  to  appreciate  the  character  of  sounds 
and,  relaxing  the  attention,  classifies  them  as  noises. 

Small  sticks  of  resonant  wood  may  be  prepared.  Fig.  23, 
such  that  when  dropped,  the  resulting  sound  is  a  mixture  of 
noise  and  simple  musical  tones.  If  several  of  these  sticks 
are  dropped  together,  the  sound  gives  the  effect  of  noise 
only,  while  if  the  sticks  are  dropped  one  at  a  time  in  proper 
order,  the  ear  clearly  distinguishes  a  musical  melody  in 
spite  of  the  accompanying  noise.  The  drawing  of  a  cork 
from  a  bottle  expands  the  contained  air ;  when  the  cork  is 
wholly  withdrawn,  the  air,  because  of  its  elasticity,  vibrates 
with  a  frequency  dependent  upon  the  size  and  shape  of 
the  bottle.  The  resulting  sound  is  of  short  duration  and 
is  thought  of  only  as  a  popping  sound,  while  it  is  in  reality 
a  musical  tone.  The  musical  characteristic  is  made  evident 
by  drawing  the  plugs  from  several  cylindrical  bottles,  Fig. 

22 


NOISE  AND  TONE 


23,  the  tones  of  which  are  in  the  relations  of  the  common 
chord,  do,  mi,  sol,  do.  A  distinguishable  tune  can  be  played 
on  a  flute  without  blowing  into  it,  the  air  in  the  tube  being 
set  in  vibration  by  snapping  the  keys  sharply  against  the 
proper  holes  to  give  the  tune. 

A  conspicuous  instance  of  the  change  in  classification  of  a 
musical  composition  from  noise  to  music  is  provided  by 
Wagner's  "  Tannhauser  Overture. After  this  overture  had 
been  known  to  the  musical  public  for  ten  years  it  was  criti- 


FiG.  23.    Sticks  and  bottles  which  produce  musical  noises. 


cized  in  the  London  Times  as  "  at  best  but  a  commonplace 
display  of  noise  and  extravagance."  A  Frankfort  (Germany) 
critic  said  in  1853  that  ^Tannhauser,'  so  far  as  the  public 
is  concerned,  may  be  considered  a  thing  of  the  past."  It 
was  called  shrill  noise  and  broken  crockery  effects." 
The  eminent  musical  pedagogue,  Moritz  Hauptmann  (1846), 
pronounced  it  ''quite  atrocious,  incredibly  awkward  in  con- 
struction, long  and  tedious.  It  seems  to  me,"  he  says, 
^'that  a  man  who  will  not  only  write  such  a  thing,  but 

23 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


actually  have  it  engraved,  has  little  call  for  an  artistic 
career." 

Sidney  Lanier,  the  poet-musician,  who  better  understood 
this  composition,  wrote  in  a  letter  to  his  wife,"^  ^'Ah,  how  they 
have  belied  Wagner !  I  heard  Thomas '  orchestra  play  his 
overture  to  '  Tannhauser.'  The  'Music  of  the  Future'  is 
surely  thy  music  and  my  music.  Each  harmony  was  a 
chorus  of  pure  aspirations.  The  sequences  flowed  along, 
one  after  another,  as  if  all  the  great  and  noble  deeds  of 
time  had  formed  a  procession  and  marched  in  review  before 
one's  ears,  instead  of  one's  eyes.  These  '  great  and  noble 
deeds '  were  not  deeds  of  war  and  statesmanship,  but 
majestic  victories  of  inner  struggles  of  a  man.  This  un- 
broken march  of  beautiful-bodied  Triumphs  irresistibly 
invites  the  soul  of  man  to  create  other  processions  like  it. 
I  would  I  might  lead  so  magnificent  a  file  of  glories  into 
heaven! " 

As  compared  with  the  usual  composition  of  its  time 
^'  Tannhauser  Overture  "  must  be  considered  as  having  a  com- 
plicated construction.  There  is  an  accompaniment,  quite 
independent  of  the  main  theme,  which  forms  a  beautiful 
background  of  tone,  upon  which  the  noble  melody  is  pro- 
jected. Many  of  the  early  listeners  may  have  given  their 
attention  to  this  accompaniment  and  so  have  lost  the  im- 
pressiveness  of  the  melody ;  to  them  it  was  a  confused  mass 
of  tone  producing  the  effect  of  noise. 

The  study  of  noises  is  essential  to  the  understanding  of 
the  qualities  of  musical  instruments,  and  especially  of  speech. 
Words  are  multiple  tones  of  great  complexity,  blended  and 
flowing,  mixed  with  essential  noises.  If  with  the  vowel 
tone  d  (mat)  we  combine  a  final  noise  represented  by  t,  the 
word  a+t  is  produced ;  if  to  this  simple  combination  we  add 

24 


NOISE  AND  TONE 


various  initial  noises,  several  words  are  formed,  as :  6+at, 
c+at,  /+at,  /i+at,  m+at,  p+at,  r+at,  s+at,  t+sit,  v+Sit.  However, 
the  study  of  noises  may  well  be  passed  until  we  understand 
the  simpler  and  more  interesting  musical  tones. 

Tones  are  sounds  having  such  continuity  and  definiteness 
that  their  characteristics  may  be  appreciated  by  the  ear, 
thus  rendering  them  useful  for  musical  purposes ;  these 
characteristics  are  pitch  or  frequency,  loudness  or  intensity ^ 
and  quality  or  tone  color. 


25 


LECTURE  II 
CHARACTERISTICS  OF  TONES 
Pitch 

The  pitch  of  a  sound  is  that  tone  characteristic  of  being 
acute  or  grave  which  determines  its  position  in  the  musical 

scale;  an  acute 
sound  is  of  high 
pitch,  a  grave  sound 
is  of  low  pitch.  Ex- 
periment proves 
that  pitch  depends 
upon  a  very  simple 
condition,  the  num- 
ber of  complete  vi- 
brations per  second  ; 
this  number  is  called 
the  frequency  of  the 
vibration. 

One  of  the  sim- 
plest methods  of  de- 
termining pitch  is 

Fig.  24.    Serrated  disk  for  demonstrating  thede-     mechanically  tO 
pendence  of  pitch  upon  frequency  of  vibration.         ^^^^^^  vibratioUS  at 

a  rate  which  is  known  and  which  can  be  varied  as  de- 
sired; the  rate  is  adjusted  until  the  resulting  sound  is 

26 


CHARACTERISTICS  OF  TONES 


in  unison  with  the  one  to  be  measured,  then  the  number 
of  vibrations  generated  by  the  machine  is  the  same  as  that 
of  the  sound. 

If  a  card  is  held  against  the  serrated  edge  of  a  revolving 
disk,  Fig.  24,  the  pulsations  of  the  card  produce  vibrations 
in  the  air,  and  give  rise  to  an  unpleasant  semi-musical 
sound,  having  a  recognizable  pitch  which  is  measured  by 
the  number  of  taps 
given  to  the  card 
per  second.  The 
four  disks  shown  in 
the  illustration 
have  numbers  of 
teeth  in  the  ratios 
of  4  :  5  : 6  :  8,  sound- 
ing the  common 
chord,  the  pitch  of 
which  is  dependent 
upon  the  rate  of  ro- 
tation of  the  disk. 

The  siren  is  an  in- 
strument in  which 
the  vibrations  are 
produced  by  inter- 
rupting a  jet  of  compressed  air  by  means  of  a  revolving 
disk  with  holes,  as  illustrated  in  Fig.  25 ;  the  sound  is 
much  softer  and  more  musical  than  that  from  the  ser- 
rated disk.  The  siren  has  been  developed  into  an  instru- 
ment suitable  for  research.  Fig.  26,  which  enables  one  in  a 
few  minutes  of  time  to  determine  to  one  part  in  a  hundred 
the  number  of  vibrations  of  common  musical  sounds.  For 
securing  greater  range  or  for  sounding  several  tones  simul- 

27 


Fig.  25.    The  siren. 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


taneously,  the  siren  is  usually  provided  with  two  disks, 
di  and  d2,  each  having  four  rows  of  holes ;  one  or  more 
rows  may  be  used  at  the  same  time,  each  producing  its 
own  pitch.^  The  disks  may  be  rotated  by  compressed  air 
on  the  principle  of  the  turbine,  or  by  an  electric  motor, 
as  shown  in  the  illustration ;  in  either  case  the  speed  can 
be  controlled,  and  the  number  of  vibrations  is  determined 
with  the  aid  of  the  revolution  counter  c  between  the  two 
disks. 


Fig.  26.    Siren  for  the  determination  of  pitch. 


There  are  various  other  methods  for  determining  and  com- 
paring the  number  of  vibrations  of  sounding  bodies,  which 
are  described  in  the  references.^  For  the  present  purpose  it 
will  be  sufficient  to  explain  those  used  for  the  more  precise 
determinations  of  a  fundamental  nature,  the  method  of 
beats,  Lissajous's  optical  method,  and  the  methods  of  the 
clock-fork  and  the  phonodeik. 

28 


CHARACTERISTICS  OF  TONES 


The  Tuning  Fork 

Perhaps  the  most  important  of  acoustical  instruments 
is  the  tuning  fork  invented  in  1711  by  John  Shore,  Handel's 
trumpeter.  The  fork  reached  an  almost  perfect  develop- 
ment under  the  exquisite  workmanship  and  painstaking 
research  of  Rudolph  Koenig  of  Paris.  When  properly  con- 
structed and  mounted,  it  gives  tones  of  great  purity  and 
constancy  of  pitch ;  it  is  of  very  great  value  in  experimental 


Fig.  27.    Timing  fork.s  of  various  types. 


work  and  provides  the  almost  universal  method  of  indicating 
and  preserving  standard  pitches  for  all  purposes.^  Fig.  27 
shows  various  forms  of  tuning  forks,  while  Fig.  42  represents 
a  larger  collection,  and  many  special  forms  are  shown  in 
other  illustrations. 

A  tuning  fork  for  scientific  purposes  should  be  made  of 
one  piece  of  cast  steel,  not  hardened ;  the  shapes  developed 
by  Koenig  have  not  been  excelled ;  the  patterns  for  forks 
of  ordinary  musical  pitches  and  those  of  very  high  pitches 
giving  loud  tones  are  shown  in  Fig.  28 ;  a  fork  for  A3  =  435, 
of  the  first  shape,  is  129  milhmeters  long,  not  including  the 

29 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


handle ;  a  fork  of  the  second  shape,  79  milUmeters  long, 
has  the  pitch  3328. 

The  number  of  vibrations  of  a  fork  is  dependent  upon  the 
mass  of  the  prongs  and  the  elastic  forces  due  principally  to 
the  yoke ;  if  the  prongs  are  made  lighter,  by  filing  on  the 
ends  or  sides,  the  pitch  is  raised  ;  if  the  fork  is  filed  near  the 
yoke,  the  elastic  restoring  force  is  diminished  and  the  pitch 
is  lowered.  The  second  shape  of  fork  shown  in  the  figure 
has  a  yoke  which  is  very  thick  in  proportion  to  the  prongs, 
hence  it  is  suitable  for  high  pitches.  A  standard  fork,  hav- 
ing been  accurately  machined  and  finished,  should  be  left 


Filing  or  grinding  a  fork  will  heat  it,  as  will  also  the  touch  of 
the  fingers ;  the  heating  lowers  the  pitch  of  the  fork,  and  if 
it  is  tuned  while  thus  heated,  it  will  later  be  found  too  sharp, 
that  is,  the  prongs  are  already  too  short.  Therefore  the 
filing  should  stop  while  the  fork  is  yet  two  or  three  tenths  of 
a  \'ibration  flat,  and  the  fork  should  be  allowed  to  remain  at 
a  uniform  temperature  for  a  day  or  two  before  a  comparison 
is  made  ;  if  further  tuning  is  necessary,  it  must  be  done  with 
extreme  care,  and  a  comparison  again  made  after  another 
interval  of  rest.  Vigorous  fiUng  will  produce  molecular 
disturbances  which  subside  only  after  long  periods  of  rest. 
The  methods  of  comparison  are  described  in  the  succeeding 
articles. 


Fig.  28.    Shapes  of  Koenig's  tuning  forks. 


with  the  prongs  a  trifle 
too  long,  that  is,  flat  in 
pitch ;  the  final  tuning 
should  be  carried  out 
very  carefully  by  short- 
ening both  prongs  to- 
gether till  the  desired 
frequency    is  secured. 


30 


CHARACTERISTICS  OF  TONES 


Tuning  forks  are  often  finished  with  a  bright  steel  surface, 
in  which  case  care  is  required  to  prevent  rust ;  smearing 
with  vaseUne  is  a  convenient  rust  preventive.  A  blued 
steel  finish  is  excellent ;  standard  forks  are  sometimes  blued 
over  the  entire  surface  after  all  machine  work  has  been 
finished,  but  before  the  final  tuning;  the  final  adjustment 
of  pitch  is  made  by  careful  grinding  on  the  ends  of  the 
prongs,  which  are  thus  made  bright,  and  the  surfaces  are 
then  very  lightly  etched  with  a  seal.  Any  further  alteration 
of  the  fork,  or  an  injury,  will  disfigure  it  and  will  be  easily 
detected. 

The  boxes  on  which  the  forks  are  commonly  mounted 
were  first  used  by  Marloye ;  they  are  of  such  dimensions 
that  they  form  resonance  chambers  not  quite  in  tune  with 
the  fork  tone ;  if  the  tuning  is  perfect,  the  sound  is  louder 
but  of  short  duration,  because  the  energy  of  the  vibration  is 
more  rapidly  dissipated.^  The  box  serves  a  double  purpose  : 
it  produces  a  louder  sound  and  it  also  purifies  the  tone  by 
reinforcing  only  the  fundamental.  When  the  resonance 
box  is  not  exactly  in  tune  with  the  fork,  it  draws  the  fork 
out  of  its  natural  frequency  by  a  small  amount,  a  few 
thousandths  of  a  vibration  per  second.  These  effects  are 
considered  at  greater  length  under  Resonance  in  Lecture  VI. 

Koenig  proved  that  change  of  temperature  alters  the 
number  of  vibrations  of  a  fork ;  the  temperature  coefficient 
was  found  to  be  nearly  constant  for  forks  of  all  pitches  and 
to  have  the  value  -  0.00011.^  The  change  in  the  number  of 
vibrations  of  a  fork  is  found  by  multiplying  its  frequency 
by  this  coefficient  and  by  the  number  of  degrees  of  tempera- 
ture change ;  the  negative  sign  means  that  the  frequency  is 
diminished  by  increased  temperature.  For  instance,  a  fork 
giving  435  \dbrations  per  second  at  15°  C.  will  have  its  fre- 

31 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


quency  diminished  by  0.48  vibration  for  ten  degrees  increase 
in  temperature. 

The  pitch  of  a  fork  changes  sUghtly  with  the  amphtude, 
that  is,  with  the  loudness  of  the  tone  which  the  fork  is  giving ; 
the  greater  the  amphtude,  the  less  the  frequency.  For 
extreme  changes  in  amplitude,  the  number  of  vibrations 
may  vary  as  much  as  one  in  three  hundred.  If  a  fork  is 
sounded  loudly,  the  pitch  will  rise  shghtly  as  the  tone  sub- 
sides ;  the  true  pitch  is  that  corresponding  to  a  small  inten- 
sity.io 

Tuning  forks  may  be  excited  by  bowing  across  the  end 
of  one  prong  with  a  violin  or  a  bass  bow ;  this  is  perhaps  the 
best  method  for  obtaining  the  loudest  possible  response. 
For  usual  experimental  work  the  most  convenient  method  is 
to  strike  the  fork  with  a  soft  hammer ;  a  felt  piano  hammer 
head  with  a  flexible  spring  handle  is  an  excellent  tool  for 
the  purpose ;  a  solid  rubber  ball  or  a  rubber  stopper  is  often 
used  for  the  hammer  head.  For  sounding  the  thick  high- 
pitched  forks,  an  ivory  hammer  is  best.  Forks  should 
never  be  struck  with  metal  or  other  hard  substances,  for 
being  of  soft  steel,  they  are  likely  to  be  injured. 

Forks  are  often  made  to  sound  continuously  by  means  of 
an  electro-magnetic  driving  arrangement ;  a  fork  may  be 
driven  by  itself,  its  own  vibrations,  once  started,  serving  to 
produce  the  interrupted  current  required;  such  a  fork  is 
shown  in  Fig.  50,  page  65.  Often  a  fork  is  driven  by  an 
interrupted,  or  by  an  alternating,  current  produced  from 
some  other  source ;  the  ten  forks  shown  in  Fig.  179  are  all 
driven  by  one  interrupter  fork  at  the  back  of  the  apparatus. 
In  this  instance  the  periods  of  the  forks  are  exact  multiples 
of  that  of  the  interrupter,  since  a  fork  will  respond  only  to 
impulses  which  are  in  step  with  its  own  natural  vibrations. 

32 


CHARACTERISTICS  OF  TONES 


When  a  fork  is  driven  by  this  method,  the  prong  is  inter- 
mittently urged  forward  by  the  magnetic  pull.  The  prong 
itself  is  always  a  very  little  behind  the  pull,  that  is,  it  lags 
more  or  less ;  this  forcing  of  the  vibration  causes  the  period 
to  be  slightly  different  from  that  of  the  same  fork  vibrating 
freely.^^ 

A  fork  retains  its  pitch  with  great  constancy ;  ordinary 
careless  handling  causes  little  change,  and  even  rust,  as  it 
slowly  proceeds  over  a  period  of  years,  produces  but  slight 
effect,  rarely  exceeding  one  vibration  in  two  hundred  and 
fifty ;  the  change  usually  flattens  the  pitch,  since  rust  near 
the  yoke  affects  the  fork  more  than  that  near  the  end  of 
the  prong.  The  ordinary  wear  on  a  fork  is  usually  greater 
at  the  ends  which  are  unprotected,  and  this  causes  the  pitch 
to  sharpen ;  rust  and  wear,  then,  in  some  degree  produce 
opposite  effects  and  tend  to  maintain  the  original  pitch. 

An  account  of  the  tone  quality  of  the  tuning  fork  is  given 
in  Lecture  VI,  while  many  illustrations  of  its  usefulness  will 
be  found  throughout  the  lectures. 

Determination  of  Pitch  by  the  Method  of  Beats 

A  simple  comparison  by  the  ear  will  enable  one  who  is 
musically  trained  to  tune  certain  intervals,  such  as  unisons, 
octaves,  thirds,  fourths,  and  fifths.  Two  tones  nearly  in 
unison  produce  heats,  the  number  of  which  per  second  is 
equal  to  the  difference  in  pitch  (see  page  183).  Beats  often 
occur  between  the  overtones  of  sounds  which  are  not  simple, 
and  under  other  conditions  which  need  not  be  considered 
here.  Comparison  by  ear,  based  on  the  method  of  beats,  is 
the  principal  means  employed  in  tuning  pianos  and  organs 
and  such  stringed  instruments  as  the  violin  and  the  guitar. 

The  comparison  of  a  standard  tuning  fork  with  an  un- 
D  33 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


known  pitch  of  nearly  the  same  frequency  can  be  made 
with  ease  and  precision  by  the  method  of  beats.  The 
unknown  sound  and  that  of  the  standard  fork  being  heard 
simultaneously,  the  number  of  beats  per  second  is  determined 
by  counting  the  number  occurring  in  five  or  ten  seconds ; 
the  required  pitch  is  then  that  of  the  standard  increased  or 
diminished  by  the  number  of  beats  per  second.  Usually 
the  ear  will  decide  whether  the  sound  is  flatter  or  sharper 
than  the  standard  ;  in  other  cases  it  may  be  possible  to  make 
an  easy  adjustment  to  assist  in  this  determination.  When 
the  sound  is  from  a  tuning  fork,  one  prong  may  be  loaded 
with  a  small  piece  of  wax,  which  will  slightly  lower  its  pitch  ; 
if  there  are  now  more  beats  per  second,  the  fork  is  flat, 
since  making  it  flatter  puts  it  further  out  of  tune,  and  vice 
versa.  The  fork  may  be  adjusted  to  equality  with  the  stand- 
ard by  filing,  as  already  explained,  till  the  beats  become 
fewer  and  finally  cease. 

When  the  two  sounds  approach  unison,  the  interval  be- 
tween beats  becomes  longer ;  when  the  beats  are  slow,  it  is 
difficult  to  measure  the  time  between  them,  for  one  is  not 
sure  of  the  instant  of  minimum  or  maximum  sound.  It  is 
found  that  one  can  count  beats  with  accuracy  at  the  rate 
of  from  two  to  five  per  second,  the  count  being  carried  over 
five  or  ten  seconds,  or  more  ;  four  beats  per  second  is  perhaps 
the  most  convenient  number.  For  these  reasons  an  auxiliary 
fork  is  often  used,  which  is  tuned  four  beats  per  second 
sharper  than  the  standard ;  the  fork  being  tested  is  then 
adjusted  till  it  is  four  beats  per  second  flatter  than  the 
auxiliary,  when  it  is,  of  course,  exactly  in  unison  with  the 
standard. 

Sets  of  forks  are  made  for  setting  or  testing  the  chromatic 
scale  of  equal  temperament,  as  in  tuning  pianos  and  organs, 

34 


CHARACTERISTICS  OF  TONES 


in  which  the  comparisons  are  made  by  beats.  A  series  of 
thirteen  forks  is  accurately  tuned  to  the  chromatic  scale 
from  middle  C  to  C  an  octave  higher;  an  auxiliary  set  of 
thirteen  forks  is  then  tuned  so  that  each  is  exactly  four  beats 
per  second  sharper  than  the  corresponding  fork  of  the  first 
series ;  a  correctly  tuned  octave  must  have  its  successive 
tones  four  beats  per  second  flatter  than  those  of  the  auxiliary 


Fig.  29.    Sets  of  forks  for  testiiifj;  the  accuracy  of  tuning  the  chromatic  scale. 


forks.  Such  forks  are  shown  in  Fig.  29 ;  for  making  the 
tests  the  auxiliary  forks  only  are  actually  required,  but  it  is 
desirable  to  have  the  others  also.  The  first  and  last  forks 
of  the  scale  set,  which  are  an  octave  apart,  give  258.65  and 
517.3  vibrations,  respectively,  for  A  =  435 ;  the  auxiliary 
forks  being  four  vibrations  sharp  give  262.65  and  521.3  vibra- 
tions, and  are  not  a  true  octave  apart ;  for  a  true  octave  the 
higher  fork  would  be  eight  vibrations  sharp  and  give  525.3 
vibrations ;  none  of  the  auxiliary  forks  gives  true  musical 
intervals. 


35 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


36 


CHARACTERISTICS  OF  TONES 


The  absolute  number  of  \'ibrations  may  be  determined  by 
counting  the  number  of  beats  between  the  successive  forks 
of  a  series  of  fifty  or  more  ranging  over  one  octave,  accord- 
ing to  the  method  de^^sed  by  Scheibler  in  1834. ^'^  Scheibler's 
tonometer  consisted  of  fifty-six  forks  having  pitches  from 
about  220  to  440,  the  successive  forks  differing  by  four  vi- 
brations per  second. 

This  method,  which  is  very  laborious,  has  been  used  by 
Ellis  and  by  Koenig.  Koenig's  masterpiece  is  perhaps  a 
tonometer  consisting  of  a  hundred  and  fifty  forks  of  exquisite 
workmanship,  and  tuned  vn.th.  the  greatest  care  and  skill ;  it 
covers  the  entire  range  of  audible  sounds  from  16  to  21,845.3 
vibrations  per  second. The  largest  fork  is  about  five  feet 
long,  and  has  a  cylindrical  resonator  eight  feet  in  length 
and  twenty  inches  in  diameter.  It  is  possible  to  find  in 
this  series  a  fork  which  shall  differ  from  any  given  musical 
tone  by  not  more  than  four  beats  per  second,  a  comparison 
with  which  by  the  method  of  beats  vrW\  determine  the  pitch 
of  the  sound  with  great  ease  and  precision. 

Optical  Comparison  of  Pitches 

One  of  the  most  precise  methods  for  the  comparison  of 
frequencies  is  Lissajous's  optical  method, which  depends 
upon  the  geometrical  figures  traced  by  two  simple  harmonic 
motions  at  right  angles.  The  motions  may  be  pro\dded 
by  tuning  forks  which  carry  mirrors  on  the  prongs,  as  shown 
in  Fig.  30.  A  ray  of  Ught  is  reflected  from  one  fork  to  the 
other  and  then  to  a  screen  or  an  observing  telescope.  When 
the  forks  are  vibrating,  the  ray  is  deflected  in  two  directions, 
so  that  the  figure  on  the  screen  corresponds  to  the  com- 
pounded motion.  The  shape  of  this  figure  is  characteristic 
of  the  ratio  of  the  frequencies  of  the  two  forks ;  for  certain 

37 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


simple  ratios,  such  as  1:2,  1:3,  2:3,  etc.,  the  figures  are 
easily  recognized  by  the  eye ;  and  when  the  ratio  is  exact, 
the  figure  exactly  retraces  itself,  and  because  of  the  per- 
sistence of  vision  it  appears  continuous  and  stationary.  If 
the  ratio  of  frequencies  is  not  exact,  the  figure  changes, 
because  of  progressive  phase  difference,  and,  passing  through 
a  cycle,  returns  to  the  original  form  ;  the  time  for  this  cyclic 
change  is  that  required  for  one  fork  to  gain  or  lose  one  com- 
plete vibration  on  the  exact  number  corresponding  to  the 
indicated  ratio.  The  application  of  this  method  is  explained 
in  connection  with  the  clock-fork. 

The  Clock-Fork 

The  most  precise  determinations  of  absolute  pitch  are 
those  made  by  Koenig,  who  investigated  the  influence  of  the 
resonance  box  and  of  temperature  on  the  frequency  of  a  stand- 
ard fork.  He  also  determined  the  frequency  of  the  forks 
used  by  the  Conservatory  of  Music  and  the  Grand  Opera 
in  Paris.^^  By  combining  the  clock-fork  of  Niaudet  with  a 
vibration  microscope  for  observing  Lissajous's  figures,^^ 
he  developed  the  beautiful  instrument  shown  in  Fig.  31. 
Fig.  31  is  reproduced  from  an  autographed  photograph  of 
the  original  instrument,  in  the  author's  possession,  while  the 
instrument  which  was  exhibited  in  the  lecture  is  of  more 
recent  construction  and  is  shown  in  Fig.  32,  on  page  40. 

The  apparatus  is  essentially  a  pendulum  clock  in  which 
the  ordinary  pendulum  is  replaced  by  a  tuning  fork;  the 
fork  has  a  frequency  of  64,  as  scientifically  defined ;  that  is, 
it  makes  128  swings  per  second,  counting  both  to  and  fro 
movements.  The  clock  has  the  usual  hour,  minute,  and 
second  hands ;  but  instead  of  the  escapement  operating  on 
the  second  hand  to  release  it  once  a  second,  the  gearing  of 

38 


charactp:ristics  of  tones 


the  movement  is  carried  one  step  higher,  and  a  fourth  hand 
is  provided,  which  goes  round  once  in  a  second.  A  very 
small  escapement  mechanism  is  attached  to  this  hand  and 
is  so  arranged  that  it  is  operated  by  one  prong  of  the  tuning 
fork  as  it  swings  to  and  fro,  128  times  a  second ;  the  fork 
thus  releases  the  wheels  regularly,  as  does  an  ordinary  pen- 
dulum, and  the  clock  ^'runs." 
Moreover,  as  in  the  common 
clock,  the  escapement  not 
only  releases  the  wheelwork, 
but  it  also  imparts  a  small 
impulse  to  the  fork  so  as  to 
maintain  its  vibration  as  long 
as  the  clock  runs,  that  is,  for 
days  if  desired.  Thus  we  have 
a  tuning  fork  which  will  vi- 
brate continuously,  and  a 
clock-work  which  accurately 
counts  the  vibrations. 

The  rate  of  the  fork  is  ad- 
justed much  as  is  a  pendu- 
lum, by  moving  small  weights 
up  or  down  on  threaded  sup- 
ports.    If  the  clock  is  regU-     ^.  31 .  Photograph  of  Koenig's  clock- 

lated  till  it  keeps  correct  time,  ^''''^  bearing  his  autograph, 

the  fork  must  vibrate  exactly  128  times  a  second,  making 
11,059,200  single  vibrations  in  a  day.  A  change  in  the  rate  of 
the  clock  of  one  second  per  day  means  a  change  in  the  fre- 
quency of  the  fork  of  one  part  in  eighty-six  thousand  four 
hundred ;  that  is,  when  the  clock  loses  one  second  a  day, 
the  fork  has  a  frequency  of  63.99926.  If  it  is  desired,  for 
instance,  to  adjust  the  fork  to  exactly  63  complete  vibra- 

39 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


tions  per  second,  the  clock  must  lose  one  part  in  sixty-four, 
that  is,  it  must  lose  22J  minutes  a  day. 

By  means  of  the  weights  the  actual  fork  can  be  adjusted 
to  have  any  desired  frequency  between  62  and  68,  this 
frequency  being  determined  to  a  ten-thousandth  part  of  a 


Fig.  32.    Clock-fork  arranged  for  verifying  another  fork. 


vibration  by  the  rate  at  which  the  clock  gains  or  loses.  The 
range  of  the  fork  is  a  musical  semi-tone ;  by  using  various 
multiples  of  its  frequency,  it  is  possible  to  determine  almost 
any  desired  musical  pitch  with  precision. 

The  method  of  using  the  clock-fork  may  be  illustrated  by 
a  concrete  example;  thus  the  verification  of  a  standard 
A  =  435  fork  requires  the  following  procedure.    The  only 

40 


CHARACTERISTICS  OF  TONES 


integral  divisor  of  435  which  will  give  a  quotient  within  the 
limits  of  frequency  of  the  clock-fork  is  7,  and  the  quotient 
is  62.143;  a  numerical  calculation  shows  that  if  the  clock 
loses  41  minutes  46  seconds  per  day,  or  1  minute  44.4  seconds 
per  hour,  the  tuning-fork  pendulum  wdll  make  62.143  vi- 
brations per  second ;  if  the  A-fork  vibrates  exactly  7  times 
as  fast,  its  frequency  must  be  435 ;  the  exact  ratio  of  the 
frequencies  is  to  be  determined  by  Lissajous's  figures  with 
the  vibration  microscope. 

The  clock-fork  carries  the  objective  lens  of  a  microscope, 
the  body  of  which  is  attached  to  the  frame.  The  A-fork 
is  supported  so  that  its  line  of  vibration 
is  at  right  angles  to  that  of  the  lens.  Fig. 
32,  and  so  that  some  brightly  illuminated 
point,  as  a  speck  of  chalk  dust  on  the  end 
of  the  prong,  is  visible  through  the  micro- 
scope ;  if  the  two  forks  are  vibrating,  this  fig.  33.  Lissajous's 
speck  is  seen  to  describe  the  Lissajous  figure  for  the  ratio 
curve  for  the  ratio  of  1:7,  Fig.  33.  Sup- 
pose now  the  figure  goes  through  its  cyclic  change  once  in  5 
seconds,  then  the  fork  has  a  frequency  of  either  434.8,  or  435.2. 
To  determine  whether  the  fork  is  sharp  or  flat  a  very  small 
piece  of  wax  is  attached  to  one  prong,  which  will  make  it  vi- 
brate more  slowly.  If  the  cyclic  change  requires  a  longer 
time  than  before,  the  shght  lowering  of  pitch  has  improved  the 
tuning,  which  condition  indicates  that  the  frequency  of  the 
fork  was  435.2  ;  if  the  change  occurs  in  less  time,  the  lowering 
of  the  pitch  has  made  it  further  from  the  true  value  and  the 
fork  had  a  frequency  of  434.8.  The  A-fork  may  be  adjusted 
by  filing  or  grinding,  near  the  ends  of  the  prongs  to  make 
it  sharper  or  near  the  yoke  to  make  it  flatter,  as  described 
on  page  30,  and  the  adjustment  may  be  continued  till  any 

41 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


required  accuracy  has  been  obtained ;  for  instance,  if  the 
cychc  change  occurs  in  10  seconds,  the  error  of  tuning  is 
o\  vibration  per  second. 

The  clock-fork  is  pro\dded  with  a  mirror  on  the  side  of 
one  prong  so  that  it  may  be  used  to  produce  Lissajous's 
figures  by  the  Hght-ray  method  or  to  record  the  vibrations 
directly  on  a  photographic  film. 


Pitch  Limits 

The  range  of  pitch  for  the  human  voice  in  singing  is  from 
60  for  a  low  bass  voice  to  about  1300  for  a  very  high  soprano. 


Fig.  34.    Organ  pipe  over  32  feet  long  giving  16  vibrations  per  second. 


The  piano  has  a  range  of  pitch  from  27.2  to  4138.4.  The 
pipe  organ  usually  has  16  for  the  lowest  pitch  and  4138  for 
the  highest ;  an  organ  pipe  giving  16  vibrations  per  second, 
Fig.  34,  is  nominally  32  feet  long,  though  its  actual  length 
is  somewhat  greater ;  there  are  a  few  organs  in  the  world 
having  pipes  64  feet  long  which  give  only  8  vibrations  per 
second,  but  such  a  sound  is  hardly  to  be  classed  as  a  musical 
tone ;  the  frequency  4138  is  given  by  a  pipe  1^  inches  long. 

Neither  speech  nor  music  makes  direct  use  of  all  the 
sounds  which  the  ear  can  hear.  Helmholtz  considered  32 
vibrations  per  second  as  the  lowest  limit  for  a  musical  sound, 
that  is,  one  which  gives  the  sensation  of  a  continuous  tone ; 
yet  the  piano  descends  to  27  and  the  organ  to  16  or  even  to 
8  vibrations  per  second.    The  tuning  fork  shown  in  Fig.  35 

42 


CHARACTERISTICS  OF  TONES 


may  be  made  to  give  from  16  to  32  vibrations  per  second, 
according  to  the  position  of  the  weights  on  the  prongs. 
Experimenters  differ  widely  as  to  the  lower  limit,  though 
nearly  all  consider  Helmholtz's  value  too  high  ;  perhaps  the 
most  trustworthy  values  are  between  12  and  20  vibrations 
per  second,  with  a  general 
consensus  of  opinion  that 
the  lower  limit  of  audibility 
for  a  musical  tone  is  16  vi- 
brations per  second.  Of 
course,  the  ear  can  hear 
vibrations  when  they  are 
fewer  in  number  than  16  per 
second,  but  they  are  heard 
as  separated  or  discontin- 
uous sounds. 

It  is  interesting  to  notice 
that  the  frequency  of  repeti- 
tion of  an  impression  to  pro- 
duce continuity  of  sensation 
for  sound  is  practically  the 
same  as  for  light.  The  per- 
sistence of  vision  is  about 

one  tenth  of  a  second,  that      ^^g-  ^S-    Large  fork  giving  from  16  to  32 

vibrations  per  second. 

is,   an  intermittent  visual 

sensation  occurring  ten  times  or  more  a  second  produces  the 
effect  of  a  continuous  sensation  ;  for  moving  pictures  the  views 
are  usually  changed  sixteen  times  a  second,  and  the  inter- 
mittent movement,  or  vibration,  at  this  rate,  gives  the  im- 
pression of  a  continuous  motion.  If  a  screen  is  illuminated 
with  a  moving-picture  projection  apparatus  in  which  there 
is  no  picture,  the  eye  perceives  a  flicker  in  the  general  il- 

43 


THE  SCIENCE  OP  MUSICAL  SOUNDS 


luniination  when  the  intermittent  shutter  of  the  machine  is 
in  operation,  unless  the  number  of  Hght  flashes  per  second 
exceeds  a  certain  value. ^'  This  value  varies  from  ten  to  fifty 
or  more  per  second,  according  to  the  intensity  of  the  light. 
Perhaps  Helmholtz's  value  of  32  for  the  lower  limit  of  a 
tone  is  the  flicker  limit  for  the  ear. 

While  the  upper  pitch  limit  for  the  musical  scale  is  about 
4138,  the  ear  can  hear  sounds  having  frequencies  of  20,000 


Fig.  36.    Small  organ  pipe  giving  15,600  vibrations  per  second. 

or  30,000,  and  even  more  in  cases  of  extreme  sensitiveness. 
Fig.  36  shows  what  is  in  form  a  regular  organ  pipe,  one  of 
the  smallest  ever  made,  and  much  too  small  to  be  used  in 
an  organ ;  the  length  of  the  pipe  which  is  effective  in  pro- 
ducing the  tone  is  indicated  by  I  in  the  figure  and  measures 
0.25  inch.  This  pipe  sounds  Bs  and  gives  15,600  complete 
vibrations  per  second,  a  sound  which  is  clearly  audible  to 
most  listeners.  Experiments  to  determine  the  upper  limit 
of  audibility  are  often  made  with  a  Galton's  whistle.  Fig.  37, 

44 


CHARACTERISTICS  OF  TONES 


an  adjustable  whistle  or  stopped  organ  pipe  of  very  small 
dimensions/  blown  by  means  of  a  rubber  pressure  bulb. 
The  whistle  can  be  set  to  various  lengths,  indicated  by  the 
graduated  scales,  giving  high-pitched  sounds  of  known 
frequency.  Another  experimental  method  of  producing 
sounds  of  high  pitch  is  by  the  longitudinal  vibration  of  short 
steel  bars,  Fig.  38.    The  bars  are  suspended  by  silk  cords, 


Fig.  37.    Adjustable  whistle  for  determining  the  frequency  of  the  highest  audible 

sound. 

and  are  struck  on  the  ends  with  a  steel  hammer,  producing 
a  clear  metallic  ringing  sound,  which  is  the  tone  desired ; 
the  pitch  of  the  sound  is  determined  by  the  length  of  the 
bar,  a  bar  52.5  millimeters  {2^^  inches)  long  giving  32,768 
vibrations  per  second. 

Perhaps  the  most  conclusive  experiments  on  audible  and 
inaudible  tones  of  the  highest  pitch  are  those  of  Koenig, 
extending  over  a  lifetime  of  investigation,^^  in  which  obser- 
vations were  made  with  tuning  forks,  transverse  vibrations 

45 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


of  rods,  longitudinal  vibrations  of  rods,  plates,  organ  pipes, 
membranes,  and  strings.  Tuning  forks,  when  properly  con- 
structed and  used,  proved  to  be  the  most  suitable  source  of 
high  tones.  Koenig  made  his  first  experiments  in  1874  when 
he  was  forty-one  years  old  and  at  that  time  was  able  to  hear 
tones  up  to  Fq^=  23,000,  which  he  considered  the  highest 
directly  audible  simple  tone  ;  he  constructed  a  set  of  forks  up 
to  Fg  =  21,845,  which  he  exhibited  at  the  Centennial  Exposi- 
tion in  Philadelphia  in  1876.    (These  forks  and  much  other 


ii^iii  ii  iiij  Ill  1  III  1 11 1 

Fig.  38.    Steel  bars  for  testing  the  liighest  audible  frequency  of  vil>ration.  ' 


interesting  acoustic  apparatus  exhibited  by  Koenig  are  now 
in  the  laboratory  of  Toronto  University.)  In  his  fifty- 
seventh  year  the  limit  of  audibility  for  Koenig  was  Eg  = 
20,480,  and  in  his  sixty-seventh  year  it  was  DgS=  18,432. 
A  set  of  Koenig  forks  for  tones  of  high  pitch  is  shown  in  Fig. 
39  ;  Koenig  has  made  a  complete  series  of  such  forks  extend- 
ing more  than  two  octaves  above  the  limit  of  audibility  to  a 
frequency  of  90,000  complete  vibrations  (180,000  motions 
to  and  fro)  per  second.  Sounds  which  are  inaudible  are 
made  evident  by  cork-dust  figures  in  a  tube.  Fig.  39 ;  the 
stationary  air  waves  produced  by  the  vibration  of  the  fork 
at  the  end  of  the  tube  cause  the  cork  dust  to  accumulate  in 

46 


CHARACTERISTICS  OF  TONES 


little  heaps,  one  in  each  half  wave  length  of  the  sound.  The 
wave  length  in  air  for  the  tone  of  90,000  frequency  is  1.9 
millimeters,  or  0.075  inch. 

Though  the  pitch  of  the  highest  note  commonly  used  in 
music  is  4138,  overtones  with  frequencies  of  10,000,  or  more, 
probably  enter  into  the  composition  of  some  of  the  sounds 
of  music  and  speech.  The  investigation  of  these  tones  of 
very  high  pitch  should  not  be  neglected ;  however,  the 


Fig.  39.    Forks  for  testing  the  highest  audible  frequency  of  vibration. 


analytical  work  discussed  is  these  lectures  in  limited  to 
pitches  of  from  about  100  to  5000. 

While  it  would  be  interesting  to  students  of  music  to  con- 
sider the  reasons  for  the  selection  of  tones  of  certain  pitches 
to  form  scales  and  chords,  it  would  lead  us  far  from  our 
present  purpose;  it  will,  however,  be  useful  to  notice  the 
location  on  the  musical  staff  of  the  octave  points  of  the 
sounds  used  in  music  and  to  explain  the  notation  which 
designates  a  given  tone.  The  musical  staff  may  be  con- 
sidered as  composed  of  eleven  lines ;  to  assist  in  identifying 

47 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Table  of  Equally  Tempered  Scale,  A3  =  435 


Ci-Co 

Co-Ci 

Cl-C2 

C6-C7 

c 

16.17 

32.33 

64.66 

129.33 

258.65 

517.31 

1034.61 

2069.22 

c « 

17.13 

34.25 

68.51 

137.02 

274.03 

548.07 

1096.13 

2192.26 

D 

18.15 

36.29 

72.58 

145.16 

290.33 

580.66 

1161.31 

2322.62 

D« 

19.22 

38.45 

76.90 

153.80 

307.59 

615.18 

1230.37 

2460.73 

E 

20.37 

40.74 

81.47 

162.94 

325.88 

651.76 

1303.53 

2607.05 

F 

21.58 

43.16 

86.31 

172.63 

345.26 

690.52 

1381.04 

2762.08 

F  t 

22.86 

45.72 

91.45 

182.89 

365.79 

731.58 

1463.16 

2926.32 

G 

24.22 

48.44 

96.89 

193.77 

387.54 

775.08 

1550.16 

3100.33 

G  « 

on  Pi  9Q 

001  1 7 

lo42.cJ4 

3284.68 

A  ..i 

27.19 

54.37 

108.75 

217.50 

435.00 

870.00 

1740.00 

3480.00 

a:« 

28.80 

57.61 

115.22 

230.43 

460.87 

921.73 

1843.47 

3686.93 

B 

30.52 

61.03 

122.07 

244.14 

488.27 

976.54 

1953.08 

3906.17 

C 

32.33 

64.66 

129.33 

258.65 

517.31 

1034.61 

2069.22 

4138.44 

the  lines,  the  middle  one  is  omitted  except  when  required 
for  a  note,  Fig.  40 ;  additional  lines  of  short  length  are  used 

C7  -0-4138 
CqO,2069  = 
€5:^1035      =  = 
=         ^10:517  ^ 

C3-e>-259  

^Ezz=Cpa:i29  ] 

Co"cr32  = 

Fig.  40.    Middle  C  and  the  several  octaves  of  the  musical  scale. 

48 


CHARACTERISTICS  OF  TONES 


to  extend  the  compass.  The  tone  called  ''middle  C"  is 
placed  on  the  line  between  the  bass  and  treble  staffs,  and  is 
designated  by  C3 ;  in  International  Pitch  this  tone  has  258.65 
vibrations  per  second ;  the  musical  compass  is  four  octaves 
upward  and  downward  from  middle  C,  the  various  octaves 
bearing  subscripts  as  shown;  all  the  tones  of  an  octave 
between  two  C's  are  designated  by  the  subscript  of  the 
lower  C ;  that  is,  G 3  is  on  the  second  Hne  of  the  treble  staff, 
and  Gi  is  on  the  lowest  hne  of  the  bass  staff,  etc. 

The  table  opposite  gives  the  pitch  numbers  for  all  the 
tones  of  the  equally  tempered  musical  scale,  based  on  Inter- 
national Pitch,  A3  =  435. 


Musical  pitch  is  usually  specified  by  giving  the  number 


C  an  octave  higher.  The  standard  of  musical  pitch  has 
varied  greatly,  even  within  the  history  of  modern  music, 
from  the  classical  pitch  of  the  time  of  Handel  and  Mozart, 
when  it  was  A  =  422,  to  the  modern  American  Concert 
pitch  of  A  =  461.6,  a  change  of  more  than  one  and  a  half 
semi-tones.  Ellis  gives  a  table  of  two  hundred  and  forty-two 
pitches,  showing  values  for  A  ranging  from  370  to  567,  that 
is,  from  F  S  to  D  of  the  modern  musical  scale.^^  The  condi- 
tions of  use  and  cause  of  changes  in  pitch  are  described  in 
the  references.  Especially  interesting  are  the  accounts  of 
the  changes  in  Philharmonic  Pitch,  that  of  the  London 
Philharmonic  Orchestra,  which  under  Sir  George  Smart, 
in  1826,  was  A  =  433,  and  under  Sir  Michael  Costa,  in  1845, 
was  raised  to  A  =  455.    In  America,^^  the  equivalent  of 


Standard  Pitches 


sometimes  it  is  given  by  ''Middle  C, 


of  vibrations  of  the  note  called  "Violin  A, 


,  or  by  the 


I,  though 


E 


49 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


this  Philharmonic  Pitch  is  often  referred  to  as  Concert  Pitch, 
and  it  has  reached  the  high  hmit  of  A  =  461.6.  Not  only 
has  the  rise  in  pitch  been  so  great  that  artists  have  refused  to 
sing  and  instrument  strings  frequently  break  under  the  strain, 
but  the  lack  of  uniformity  also  causes  great  confusion  and 
trouble. 

A  convention  of  physicists  in  Stuttgart  in  1834  adopted 
Scheibler's  pitch  of  A  =  440,  which  has  been  much  used  in 
Germany ;  this  is  perhaps  the  first  standard  pitch. 

As  a  result  of  Koenig's  researches  with  the  clock-fork, 
the  French  Diapason  Normal,"  A=  435  at  the  tempera- 
ture of  20°  C,  was  established  in  1859.  This  was  adopted 
by  several  of  the  leading  symphony  and  opera  orchestras ; 
the  Boston  Symphony  Orchestra  adopted  this  pitch  upon 
its  organization  in  1883. 

A  committee  of  the  Piano  Manufacturers'  Association  of 
America,  of  which  General  Levi  K.  Fuller  was  chairman, 
made  an  extensive  investigation  of  musical  pitch,  assisted 
by  Professor  Charles  R.  Cross  of  Massachusetts  Institute 
of  Technology.  After  consultation  with  many  authorities 
in  this  country  and  Europe,  the  Committee,  in  1891,  adopted 
as  the  standard  the  Diapason  Normal  as  determined  by 
Koenig  and  named  it  ''International  Pitch,  A=  435,'^  at 
a  temperature  of  20°  C.  (68°  F.).  This  is  often  called  Low 
Pitch  in  distinction  from  Concert  or  Philharmonic  Pitch, 
which  is  now  referred  to  as  High  Pitch.  The  committee 
selected  as  its  fundamental  standard  the  type  of  fork  made 
by  Koenig,  shown  in  Fig.  41,  which  is  provided  with  an 
adjustable  cyHndrical  resonator  and  gives  a  tone  of  great 
strength  and  purity. 

It  has  been  proposed  that  A  =  438  be  made  a  standard, 
as  a  compromise  between  the  Stuttgart  A  =  440  and  the 

50 


CHARACTERISTICS  OF  TONES 


Diapason  Normal  A  =  435 ;  for  practical  purposes  there  is 
little  difference  in  the  pitches  435,  438,  and  440 ;  but  there 
should  be  but  one  nominal  standard,  and  it  seems  that  the 
strongest  arguments  favor  the  universal  adoption  of  A  =  435. 
The  musician  should  insist  that  his  piano  and  other  instru- 
ments be  tuned  to  this  pitch. 
_ 


Fig.  41.    Standard  fork.    International  Pitch,  A  =  435. 

Before  any  standard  had  been  generally  established  for 
musical  purposes,  Koenig  adopted  one  for  his  own  work, 
and  as  tuning  forks  of  his  make  are  widely  used  in  scientific 
institutions,  this  pitch,  in  which  middle  C  =  256,  is  often 
referred  to  as  Scientific  or  Philosophical  Pitch. 

The  author  urges  the  use  of  one  pitch  only  for  both  scien- 
tific and  musical  purposes,  viz.  A  =  435 ;  in  the  tempered 

51 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


CHARACTERISTICS  OF  TONES 


musical  scale  this  gives  for  middle  C  258.65  vibrations  per 
second.  This  pitch  is  used  exclusively  in  discussing  the 
results  of  our  sound  analysis.  In  the  laboratory  of  Case 
School  of  Applied  Science  the  scale  forks  based  on  C  =  256 
have  been  duplicated  with  new  forks  based  on  A  =  435 ; 
Fig.  42  shows  the  larger  part  of  this  collection,  there  being 
over  two  hundred  forks  in  the  picture. 

Intensity  and  Loudness 

The  loudness  of  a  sound  is  a  comparative  statement  of 
the  strength  of  the  sensation  received  through  the  ear.  It  is 
impossible  to  state  simply  the  factors  determining  loudness. 
For  the  corresponding  characteristic  of  light  (illumination) 
there  is  a  moderately  definite  standard,  commonly  called 
the  candle  power ;  but  for  sound  there  is  no  available  unit 
of  loudness,  and  we  are  dependent  on  the  subjective  com- 
parison of  our  sensations. Not  only  are  the  ears  of  differ- 
ent hearers  of  different  sensitiveness,  but  each  individual  ear 
has  a  varying  sensitiveness  to  sounds  of  different  pitches 
and,  therefore,  to  sounds  of  various  tone  colors. 

In  a  first  study  of  the  physical  characteristics  of  sounds 
we  are  compelled  to  consider  the  intensity  not  as  the  loudness 
perceived  by  the  ear,  but  as  determined  by  what  the  physi- 
cist calls  the  energy  of  the  vibration.  Fortunately,  under 
simple  conditions  and  within  the  range  of  pitch  of  the  more 
common  sounds  of  speech  and  music,  there  is  a  reasonable 
correspondence  between  loudness  and  energy. 

The  energy,  or  what  we  will  call  the  intensity  of  a  simple 
vibratory  motion,  varies  as  the  square  of  the  amplitude, 
the  frequency  remaining  constant ;  it  varies  as  the  square 
of  the  frequency,  the  amplitude  remaining  constant ;  when 
both  amplitude  and  frequency  vary,  the  intensity  varies 

53 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


as  the  square  of  the  product  of  ampHtude  and  frequency; 
or  to  express  it  by  a  formula,  representing  intensity  by  /, 
amphtude  by  A,  and  frequency  by  n, 

I  =  v}A\ 

Since  we  are  to  study  sounds  by  means  of  representative 
curves  or  wave  hues,  we  may  give  attention  to  the  features 
of  the  curves  which  indicate  intensity.    In  Fig.  43  the  curve 


n=l    A  =  \  1=1 


n=3.3y4  =  0.3  /=  1 

d  vAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 

Fig.  43.    Curves  representing  simple  sounds  of  various  degrees  of  loudness. 


h  has  a  frequency  the  same  as  that  of  curve  a,  but  its  am- 
phtude is  twice  as  great,  hence  it  represents  a  sound  four 
times  as  loud  ;  the  curve  c  has  an  amplitude  the  same  as  that 
of  a,  but  its  frequency  is  twice  as  great,  and  again  its  loud- 
ness is  four  times  that  of  a  ;  the  curve  d  has  a  frequency  of 
3.3  and  an  amplitude  of  0.3,  and  it  represents  a  loudness 
equal  to  that  of  a.  Then  the  sounds  represented  by  a  and 
d  are  of  equal  loudness ;  and  those  represented  by  b  and  c 
are  equal,  but  are  four  times  as  loud  as  a  or  d. 

54 


CHARACTERISTICS  OF  TONES 


Caution  is  necessary  when  making  inferences  from  simple 
inspection  of  photographic  records  of  sound  vibrations,  since 
a  change  of  film  speed  may  give  an  apparent  change  of 
frequency  when  none  really  exists. 

When  we  are  studying  the  records  of  complex  sounds,  and 


n  =1    A=\  1=1 


a 


n  =  10    A  =  0.2>    1=  Q 


Fig.  44.    Curves  representing  two  simple  sounds  and  their  combination. 


practically  all  sounds  are  such,  a  simple  measurement  of  the 
amplitude  of  the  curve  and  of  the  frequency  is  not  sufficient 
for  a  determination  of  the  loudness  ;  it  is  necessary  to  analyze 
the  wave  into  its  simple  components,  to  compute  the  in- 
tensity due  to  each  component  singly,  and  then  to  take  the 
sum  of  these  intensities ;  Fig.  44  illustrates  this  condition. 
Curves  a  and  b  have  loudnesses  represented  by  1  and  9, 
as  explained  above ;  curve  c  contains  both  a  and  h  and  its 
true  loudness  is  therefore  10.    If  it  w^ere  assumed  that  the 

55 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


loudness  of  a  and  c  are  represented  by  the  squares  of  their 
measured  widths,  the  value  for  c  would  be  1.6  as  compared 
with  a,  which  is  only  one  sixth  of  its  real  loudness. 

A  further  illustration  of  the  necessity  for  analysis  of  a 
wave  before  judging  of  the  loudness  is  shown  in  Fig.  45,  in 
which  a  and  h  are  of  exactly  the  same  loudness  though  of 
different  widths.  The  ciu've  a  is  composed  of  two  partials, 
a  fundamental  and  its  second  overtone,  of  loudness  1  and  4, 
respectively ;  h  is  composed  of  the  same  partials,  and  there- 


FiG.  45.  Curves  representing  the  combination  of  two  simple  sounds  in  different 
phases ;  though  the  curves  are  of  different  widths,  they  represent  sounds  of 
the  same  loudness. 

fore  has  the  same  loudness.  The  curves  differ  only  in  the 
relation  of  the  phases  of  the  components. 

Acoustic  Properties  of  Auditoriums 

The  loudness  of  a  sound  as  perceived  by  the  ear  depends 
not  only  upon  the  characteristics  of  the  source,  but  also 
upon  the  pecuharities  of  the  surroundings.  Among  the 
features  of  an  auditorium  which  must  be  considered  are  its 
size  and  shape,  the  materials  of  which  it  is  constructed, 
its  furnishings,  including  the  audience,  and  the  position 
of  the  source. 

56 


CHARACTERISTICS  OF  TONES 


The  determination  of  the  acoustic  properties  of  audi- 
toriums is  of  the  very  greatest  practical  importance,  and  it  is 
also  one  of  the  most  elusive  of  problems ;  the  sounds  which 
most  interest  us  are  of  short  duration  and  they  leave  no 
trace,  and  the  conditions  affecting  the  production,  the 
transmission,  and  the  perception  of  sound  are  extremely 
complicated.  The  difficulties  of  the  work  are  such  as  to  dis- 
courage any  but  the  most  skillful  and  determined  investi- 
gator. Indeed,  the  problem  has  been  almost  universally 
considered  impossible  of  solution  ;  and  this  opinion  has  been 
accepted  with  so  much  complacence,  and  even  with  satis- 
faction, that  it  still  persists  in  spite  of  the  fact  that  a 
scientific  method  of  determining  the  acoustic  properties  of 
auditoriums  has  been  developed  by  Professor  Wallace  C. 
Sabine  of  Harvard  University.  This  method,  which  is  of 
remarkable  practical  utility,  has  been  described  in  archi- 
tectural and  scientific  journals.--  No  auditorium,  large  or 
small,  and  no  music  room,  public  or  private,  should  be 
constructed  which  is  not  designed  in  accordance  with  these 
principles.  Sabine's  experiments  have  shown  that  the  most 
common  defect  of  auditoriums  is  due  to  reverberation,  a 
confusion  and  diffusion  of  sound  throughout  the  room  which 
obscures  portions  of  speech.  There  are  other  effects,  due 
to  echoes,  interferences,  and  reflection  in  general,  all  of  which 
have  been  considered.  In  many  cases  these  troubles  can 
be  renledied,  with  more  or  less  difficulty,  in  auditoriums 
already  constructed ;  this  is  especially  true  in  regard  to 
reverberation,  which  is  reduced  by  the  proper  use  of  thick 
absorbing  felt  placed  on  the  side  walls  and  ceihng. 

A  method  for  photographing  the  progress  of  sound  waves 
in  an  auditorium  is  referred  to  in  Lecture  III,  page  88, 
which  bears  indirectly  upon  the  loudness  of  the  sound  and 

57 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


is  of  great  value  in  designing  rooms  which  shall  be  free  from 
defects. 

A  soundboard  placed  behind  the  speaker  may,  in  some  in- 
stances, distribute  the  sound  in  such  a  way  as  to  remedy 
certain  defects,  as  has  been  shown  by  the  elaborate  experi- 
ments of  Professor  Floyd  R.  Watson, but  the  more  common 
faults  are  not  removed  by  this  method.  An  auditorium  has 
been  described  by  Professor  Frank  P.  Whitman,  which  was 
practically  unimproved  by  the  use  of  a  soundboard,  and 
was  later  made  altogether  satisfactory  for  public  speaking 
upon  the  removal  of  reverberation  by  Sabine's  method.^^ 

It  may  be  added  that  the  stringing  of  wires  or  cords  across 
an  auditorium  can  in  no  degree  whatever  remove  acoustical 
defects. 

Tone  Quality 

The  third  property  of  tone  is  much  the  most  complicated  ; 
it  is  that  characteristic  of  sounds,  produced  by  some  par- 
ticular instrument  or  voice,  by  which  they  are  distinguished 
from  sounds  of  the  same  loudness  and  pitch,  produced  by 
other  instruments  or  voices.  This  characteristic  may  be 
called  tone  color,  tone  quality,  or  simply  quality. 

With  comparatively  httle  practice  one  can  acquire  the 
ability  to  recognize  with  ease  any  one  of  a  series  of  musical 
instruments,  when  they  produce  tones  of  the  same  loudness 
and  pitch.  There  is  an  almost  infinite  variety  of  tone 
quality ;  not  only  do  different  instruments  have  character- 
istic qualities,  but  individual  instruments  of  the  same  fam- 
ily show  delicate  shades  of  tone  quality ;  and  even  notes 
of  the  same  pitch  can  be  sounded  on  a  single  instrument 
with  qualitative  variations.  The  bowed  instruments  of 
the  violin  family  possess  this  property  in  a  marked  degree. 

No  musical  instrument  equals  the  human  voice  in  the 

58 


CHARACTERISTICS  OF  TONES 


ability  to  produce  sounds  of  varied  qualities ;  the  different 
vowels  are  tones,  each  of  a  distinct  musical  quality.  The 
investigation  of  tone  quality  therefore  leads  to  a  study  of 
vocal  as  well  as  instrumental  sounds. 

Since  pitch  depends  upon  frequency,  and  loudness  upon 
amplitude  (and  frequency),  we  conclude  that  quality  must 
depend  upon  the 
only  other  prop- 
erty of  a  periodic 
vibration,  the 
peculiar  kind  or 
form  of  the  mo- 
tion ;  or  if  we 
represent  the 
vibration  by  a 
curve  or  wave 
line,  quality  is 
dependent  upon 
the  peculiarities 
represented  by 
the  shape  of  the 
curve. 

The  simplest 
possible  type  of 


Fig.  4G. 


Models  of  three  simple  waves,  having  fre- 
quencies in  the  ratios  of  1  :  2  :  3. 


vibration,  simple  harmonic  motion,  and  its  representative 
curve,  the  sine  curve,  were  described  in  the  preceding  lecture. 
A  tuning  fork,  when  properly  mounted  on  a  resonance  box, 
gives  to  the  air  a  single  simple  harmonic  motion,  w^hich, 
being  propagated,  develops  a  simple  wave.  The  sensation 
of  such  a  tone  is  absolutely  simple  and  pure. 

The  nature  of  tone  quality  may  be  explained  with  the  aid 
of  tuning  forks  and  the  wave  models     show^n  in  Fig.  46. 

59 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Let  one  of  the  forks  having  the  pitch  C3  be  sounded; 
it  will  produce  a  simple  wave  in  the  air,  which  may  be  repre- 
sented by  the  model  A  ;  a  second  fork,  one  octave  higher, 
will,  when  sounding  alone,  send  out  twice  as  many  vibrations 
per  second,  generating  simple  waves  of  just  half  the  w^ave 
length,  as  represented  by  the  model  B  ;  a  third  fork,  vibrating 

three  times  as 
fast  as  the  first, 
produces  waves 
one  third  as  long, 
shown  by  model 
C.  These  sim- 
ple models  illus- 
trate two  char- 
acteristics of 
tone :  pitch,  by 
the  frequency  or 
number  of  waves 
in  a  given  length, 
and  loudness,  by 
the  height  or 
amplitude  of 

Fig.  47.    Wave  form  resulting  from  the  composition  of      ,  i 

two  simple  waves.  WaveS. 

If  two  forks 

are  sounded  at  the  same  time,  the  two  corresponding 
simple  motions  must  exist  simultaneously  in  the  air, 
and  the  motion  of  a  single  particle  at  any  instant  must 
be  the  algebraic  sum  of  the  motions  due  to  each  fork  sepa- 
rately. This  condition  is  shown  in  Fig.  47,  where  the  wave 
B  has  been  lowered  to  rest  on  the  top  of  A,  impressing  the 
form  of  A  upon  B,  which  now  exhibits  the  form  of  the  mo- 
tion due  to  the  two  simple  sounds.    When  the  three  forks 

60 


CHARACTERISTICS  OF  TONES 


are  sounding,  the  form  of  the  composite  motion  is  shown  by 
lowering  the  wave  form  C  upon  that  of  A  and  B,  as  shown 
in  Fig.  48. 

The  relative  phase  of  a  wave  may  be  shifted  by  changing 
the  position  of  one  of  the  forks  in  relation  to  the  others; 
this  effect  is  demonstrated  by  shifting  the  corresponding 
wave  form  side- 
wise  (in  the  direc- 
tion of  the  length 
of  the  wave)  be- 
fore the  forms  are 
pushed  together ; 
the  shape  of  the 
resulting  wave  is 
thus  changed 
while  its  compo- 
sition remains  the 
same. 

This  argument 
may  be  extended 
indefinitely  to  in- 
clude any  num- 


n   X  iBDiiHiintiiii 

"   

li 

i  j 

Fig.  4h.  Wave  form  resulting  from  the  composition  of 
three  simple  waves,  corresponding  to  a  composite 
sound  containing  three  partials. 


ber  of  simple 
tones  of  any  se- 
lected frequencies, 
amplitudes,  and  phases.  There  are  therefore  peculiarities 
in  the  motion  of  a  single  particle  of  air  which  differ  for  a 
single  tone  and  for  a  combination  of  tones ;  and  in  fact  the 
kind  of  motion  during  any  one  period  may  be  of  infinite 
variety,  corresponding  to  all  possible  tone  qualities.  These 
lectures  are  concerned  almost  wholly  Avith  the  development 
and  the  application  of  this  principle. 

61 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Law  of  Tone  Quality 

The  law  of  tone  quality  was  first  definitely  stated  in  1843 
by  Ohm  of  Munich,  in  Ohm^s  Law  of  Acoustics,  and  much 
of  Helmholtz's  work  of  thirty  years  later  was  devoted  to  the 
elaboration  and  justification  of  this  law.^® 

The  law  states  :  all  musical  tones  are  periodic  ;  the  human 
ear  perceives  pendular  vibrations  alone  as  simple  tones ; 
all  varieties  of  tone  quality  are  due  to  particular  combina- 
tions of  a  larger  or  smaller  number  of  simple  tones ;  every 
motion  of  the  air  which  corresponds  to  a  complex  musical 
tone  or  to  a  composite  mass  of  musical  tones  is  capable  of 
being  analyzed  into  a  sum  of  simple  pendular  vibrations,  and 
to  each  simple  vibration  corresponds  a  simple  tone  which 
the  ear  may  hear. 

From  this  principle  it  follows  that  nearly  all  the  sounds 
which  we  study  are  composites.  The  separate  component 
tones  are  called  partial  tones,  or  simply  partials ;  the  partial 
having  the  lowest  frequency  is  the  fundamental,  while  the 
others  are  overtones.  It  sometimes  happens  that  a  partial 
not  the  lowest  in  frequency  is  so  predominant  that  it  may 
be  mistaken  for  the  fundamental,  as  wdth  bells ;  and  some- 
times the  pitch  is  characterized  by  a  subjective  beat-tone 
fundamental  when  no  physical  tone  of  this  pitch  exists.  If 
the  overtones  have  frequencies  which  are  exact  multiples  of 
that  of  the  fundamental  they  are  often  called  harmonics, 
otherwise  they  may  be  designated  as  inharmonic  partials. 

As  the  result  of  elaborate  investigation,  Helmholtz  added 
the  following  law :  the  quality  of  a  musical  tone  depends 
solely  on  the  number  and  relative  strength  of  its  partial 
simple  tones,  and  in  no  respect  on  their  differences  of  phase.^^ 
Koenig,  after  experimenting  with  the  wave  siren  (Fig.  178, 

62 


CHARACTERISTICS  OF  TONES 


page  245),  argued  that  phase  relations  do  affect  tone  quahty 
in  some  degree.-^  Lindig  has  used  a  '^telephone-siren"  and 
concludes  that  the  phases  of  the  components  influence 
quahty  of  tone  only  through  interference  effects.-^  Lloyd 
and  Agnew,  using  special  alternating  current  generators 
in  connection  with  a  telephone  receiv^er,  have  found  that 
the  phase  differences  of  the  components  do  not  affect  the 
quality  of  tone.^°  The  question  has  been  extensively  in- 
vestigated by  many  others,  vdth  sl  consensus  of  opinion  that 
Helmholtz's  statement  is  justified.^^ 

In  the  analysis  of  sound  waves  from  instruments  and 
voices,  described  in  Lectures  VI  and  VII,  the  phases  of  all 
component  tones  have  been  determined.  While  systematic 
study  of  the  phases  has  not  yet  been  made,  no  evidence  has 
appeared  which  indicates  that  the  phase  relation  of  the 
partials  has  any  effect  upon  the  quality  of  the  tone.  If  tone 
quality  varies  wdth  phase  relations,  the  variations  'certainly 
are  very  small  in  comparison  wdth  those  due  to  other  influ- 
ences. 

The  analyses  which  have  been  made  give  abundant  e\ddence 
that  tone  quality  as  perceived  by  the  ear  is  much  influenced 
by  subjective  beat-tones.  While  these  tones  may  be  con- 
sidered as  ha\dng  no  physical  existence,  yet  their  effects 
upon  the  ear  are  those  of  real  partials,  and  the  laws  already 
stated  include  them.  An  explanation  of  beat-tones  is  given 
in  Lecture  VI,  page  183. 

Fig.  49  shows  on  the  musical  staff  the  relations  of  a  funda- 
mental tone,  C2  =  129,  and  nineteen  of  its  harmonic  over- 
tones. The  numerals  in  the  line  below  the  staff  indicate 
the  orders  of  the  several  partials.  In  the  next  lower  line 
are  given  the  frequencies  of  the  partials  when  they  are 
harmonic.    Every  sound  which  is  represented  by  a  periodic 

63 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


wave  form  must  have  harmonic  overtones,  as  will  be  more 
fully  explained  in  Lecture  IV;  such  sounds  are  generally 
described  as  musical.  The  partial  tones  of  sounds  such  as 
the  clang  of  a  bell  are  inharmonic  and  would  not  correspond 
to  the  scheme  shown  in  the  figure.  The  tones  of  the  musical 
chromatic  scale  are  determined  according  to  the  scheme  of 
equal  temperament  developed  by  Bach.  The  various  har- 
monic overtones  of  a  given  sound  are  not  in  tune  with  any 
notes  of  the  musical  scale,  except  such  as  are  one  or  more 


12    3    4    5    6    7    8    9   10   II    12  13   14   15    16  17   18  19  20 
129  259  388  517  647  776  905  1035 1164  1293  1423  1552  1681  f8l!  1940  2069  2199  23282457  2586 

C  C  G  C  E  G  B''C  D  E  G^G  G*B''B  C  C^D  D^E 

129  259  388  517  652  775  322  1035  1161  1304  1463  1550 1642 1843  1953  2069  2192  2323  2461  2&)7 
Fig.  49.    A  fundamental  and  its  harmonic  overtones. 

exact  octaves  from  the  fundamental.  The  notes  on  the 
staff  in  Fig.  49  represent  the  scale  tones  which  are  nearest 
to  the  overtones  ;  the  lower  lines  in  the  figure  give  the  desig- 
nations of  the  notes  and  their  frequencies  in  the  tempered 
scale. 

Overtones  can  be  illustrated  by  vibrating  strings  in  such 
a  way  as  to  make  their  nature  directly  visible.  A  silk  cord 
may  be  made  to  vibrate  by  a  large  electrically  driven  fork,  as 
shown  in  Fig.  50,  with  the  formation  of  a  single  loop  due  to 
vibration  in  the  fundamental  mode.  By  changing  the  ten- 
sion of  the  string,  it  can  be  made  to  vibrate  in  various  sub- 

64 


CHARACTERISTICS  OF  TONES 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Fig.  51.    Simple  vibrations  of  a  string  in  various  subdivisions,  corresponding  to 
harmonic  overtones  or  partials. 


Fig.  52.    Complex  vibrations  of  a  string,  showing  the  coexistence  of  several  modes 
of  vibration,  representing  tones  having  different  qualities. 

66 


CHARACTERISTICS  OF  TONES 


divisions  corresponding  to  its  harmonic  overtones ;  Fig.  51 
shows  two-loop,  three-loop,  and  five-loop  formations,  repre- 
senting the  first,  second,  and  fourth  overtones.  A  string 
vibrating  in  these  forms  would  emit  simple  tones  only  ;  if  the 
pitch  for  the  single  loop  is  C2  =  129,  the  two  loops  would 
correspond  to  the  tone  C3  =  259,  the  three  loops  to  G3  = 
387,  and  the  five  loops  to  E4  =  645. 

A  string  may  be  made  to  vibrate  in  complex  modes,  with 
the  simultaneous  existence  of  several  loop  formations. 
Fig.  52  shows  the  vibrations  with  two  and  four  loops,  with 


4 

4 

5 

^  1 

5 

1 

5 

4 

5 

5 

0  m 

5 

4 

4 

4 

f-  1 

4 

4 

p 

• 

4 

•  1 

•  1 

- 

4 

•  1 

m 

»  . 

• 

9-  ' 

Fig.  53.    A  tune  in  harmonics. 


three  and  six  loops,  and  at  the  top  a  much  more  complex 
combination ;  these  forms  represent  composite  sounds,  each 
set  of  loops  corresponding  to  a  partial  tone. 

The  multiplicity  of  tones  from  one  air  column,  correspond- 
ing to  the  several  loop  formations  in  a  vibrating  string,  are 
illustrated  by  wind  instruments,  many  of  which  use  har- 
monic tones  in  their  regular  scales.  The  bugle  can  sound 
only  tones  due  to  the  vibration  of  the  air  column  in  various 

67 


THE  SCIENCE  OF  :\1USICAL  SOUNDS 


subdivisions  of  its  fundamental  length ;  it  produces  the 
tones  of  the  harmonic  series  shown  in  Fig.  49.  A  flute  tube 
without  holes  or  keys  may  be  made  to  sound  ten  or  more 
tones  of  the  harmonic  series.  A  tune  can  be  played  on  a 
flute  by  using  the  harmonic  tones  of  only  three  fundamen- 
tals, requiring  two  keys  which  are  manipulated  by  one  finger ; 
the  illustration,  Fig.  53,  shows  at  the  bottom  the  notes 
fingered,  while  those  at  the  top  are  the  harmonic  tones 
sounded ;  the  small  numerals  indicate  the  orders  of  the 
partials  used  for  the  several  tones. 

Analysis  by  the  Ear 

Even  after  the  arguments  presented,  it  may  seem  strange 
that  a  single  source  of  sound  can  emit  several  distinct  tones 
simultaneously.  There  is,  however,  abundant  experimen- 
tal evidence  in  support  of  the  statement.  By  listening 
attentively,  one  can  often  distinguish  several  component 
tones  in  the  sound  from  a  flute  or  violin  or  other  instrument. 


Fig.  54.    Helmholtz  resonators. 

Helmholtz,  who  depended  mainly  upon  the  ear  for  the 
analysis  of  composite  sounds,  developed  several  methods  for 
assisting  the  ear  in  the  detection  of  partial  tones.^^  He 

68 


CHARACTERISTICS  OF  TONES 


devised  the  tuned  spherical  resonator  which  he  used  with 
remarkable  success.  Fig.  54  shows  a  series  of  Helmholtz 
resonators  for  the  first  nineteen  overtones  of  a  fundamental 
having  a  frequency  of  64  vibrations  per  second ;  the  ten 
odd-numbered  resonators  in  the  series  correspond  to  a  fun- 
damental of  128  vibrations  per  second  and  its  first  nine  over- 
tones. The  resonator  consists  of  a  spherical  shell  of  metal 
or  glass ;  there  is  a  conical  protuberance  ending  in  a  small 
aperture,  which  is  to  be  inserted  in  the  ear ;  opposite  this 
aperture  is  an  opening,  through  which  the  sound  waves 
influence  the  air  in  the  resonator.  The  tuning  depends  upon 
the  volume  of  air  in  the  resonator  and  the  size  of  the  opening. 
If  one  ear  is  stopped  while  a  resonator  is  applied  to  the  other, 
most  of  the  tones  existing  in  the  surrounding  air  will  be 
damped  or,  in  effect,  excluded,  while  if  a  component  sound 
exists  which  is  of  the  same  pitch  as  that  of  the  resonator, 
this  particular  simple  tone  affects  the  ear  powerfully. 


69 


LECTURE  III 


METHODS  OF  RECORDING  AND  PHOTOGRAPHING 
SOUND  WAVES 

The  Diaphragm 

An  adequate  investigation  of  the  most  interesting  char- 
acteristic of  sound,  tone  quaUty,  requires  consideration  of 
the  form  of  the  sound  wave ;  for  this  purpose  it  is  desirable 
to  have  visible  records  of  the  sounds  from  various  sources 
which  can  be  quantitatively  examined  and  preserved  for 
comparative  study. 

Nearly  all  the  methods  which  have  been  developed  for 
recording  sound  make  use  of  a  diaphragm  as  the  sensitive 
receiver.  A  diaphragm  is  a  thin  sheet  or  plate  of  elastic 
material,  usually  circular  in  shape,  and  supported  more  or 
less  firmly  at  the  circumference.  The  telephone  has  a 
diaphragm  of  sheet  iron ;  in  the  talking  machine  sheets  of" 
mica  are  often  used,  while  the  soundboard  of  a  piano  is  a 
wooden  diaphragm ;  many  other  materials  may  serve  for 
special  purposes,  such  as  paper,  parchment,  animal  tissue, 
rubber,  gelatin,  soap  film,  metals,  and  glass. 

Diaphragms  respond  with  remarkable  facility  to  tones  of 
a  wide  range  of  pitch  and  to  a  great  variety  of  tone  combina- 
tions. The  telephone  transmitter,  the  recording  talking 
machine,  and  the  eardrum  illustrate  the  diaphragm  set  in 
vibration  by  the  direct  action  of  air  waves ;  one  readily 
thinks  of  the  diaphragm  as  being  affected  by  the  variations 

70 


RECORDING  AND  PHOTOGRAPHING  SOUND  WA\  ES 


in  air  pressure  which  constitute  the  wave,  but  it  is  difficult 
to  reahze  how  the  movements  can  accurately  correspond  to 
the  composite  harmonic  motion  which  represents  the  par- 
ticular tone  color  of  a  given  voice  or  instrument.  However, 
the  reproductions  of  the  telephone  and  talking  machine 
are  convincing  e\'idence  that  the  diaphragm  does  so  respond, 
at  least  to  the  degree  of  perfection  attained  by  these  instru- 
ments. 

Not  only  may  sound  waves  cause  a  diaphragm  to  vibrate, 
but  what  is  even  more  wonderful,  a  diaphragm  \dbrating 
in  any  manner  may  set  up  sound  waves  in  the  air ;  this 
reverse  action  of  the  diaphragm  is  shown  in  the  receiving 
telephone,  magnetism  being  the  exciting  cause,  and  in  the 
machine  which  talks,  the  diaphragm  of  which  is  mechanically 
pulled  and  pushed  by  the  record.  The  head  of  a  drum  is 
a  diaphragm  excited  by  percussion,  the  soundboard  of  a 
piano  is  caused  to  vibrate  by  the  action  of  the  strings,  and 
the  vocal  chords  may  be  considered  as  a  diaphragm  set  in 
vibration  by  a  current  of  air. 

The  usefulness  of  the  diaphragm  is  limited,  and  some- 
times annulled,  for  both  scientific  and  practical  purposes, 
by  certain  peculiarities  in  its  action  related  to  what  are 
called  its  natural  periods  of  \dbration ;  these  effects  of  the 
diaphragm  are  considered  in  Lecture  V. 

Various  instruments  employing  the  diaphragm,  which 
have  been  useful  in  research  on  sound  waves,  wlW  be  de- 
scribed in  the  succeeding  articles. 

The  Phoxautograph 

The  Scott-Koenig  phonautograph,  by  which  sound  waves 
are  directly  recorded,^^  was  perfected  in  1859.  The  instru- 
ment consists  of  a  membrane  placed  at  the  focus  of  a  para- 

71 


THE  SCIENCE  OF  MUSICAL  SOUNDS 

bolic  receiver  or  sound  reflector,  Fig.  55 ;  a  stylus  attached 
to  the  membrane  makes  a  trace  on  smoked  paper  carried 


Fig.  55.    Koenig's  phonautograph  for  recording  sounds. 


on  a  rotating  cyhnder ;  a  sound  produced  in  front  of  the 
receiver  causes  movements  of  the  membrane  which  are 


Fig.  56.    Phonautograph  records. 


recorded.  A  tuning  fork  with  its  prongs  between  the  mem- 
brane and  the  paper  is  mounted  on  the  base  of  the  instru- 

72 


RECORDING  AND  PHOTOGRAPIIIXG  SOUND  WAVES 


ment ;  a  stylus  attached  to  one  prong  of  the  fork  marks  a  sim- 
ple wave  line  by  the  side  of  the  trace  from  the  membrane. 

Phonautograph  records  obtained  by  Koenig  are  shown  in 
Fig.  56,  the  lower  one  of  each  pair  of  traces  is  that  of  the 
somid  being  studied,  combinations  of  organ  pipes  in  this 
instance,  while  the  upper  trace  of  each  pair  is  trom  the 
tuning  fork,  enabUng  the  determination  of  the  frequencies 
of  the  recorded  tones.  These  records  are  not  only  small 
in  size,  but  the  essential  characteristics  are  distorted  or 
obliterated  by  friction  and  by  the  momentum  of  the  st^^lus. 

The  ]\Iaxometric  Flame 

In  1862  Koenig  de\ised  the  manometric  capsule  in  which 
the  flame  of  a  burning  gas  jet  \'ibrates  in  response  to  the 
variations  in  pressm-e  in  a  sound  wave.^^  The  capsule  c, 
Fig.  57,  is  di^dded  into  two  compartments  by  a  partition 


Fig.  57.    Koenig's  manometric  capsule  with  revolving  mirror. 
73 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


of  thin  rubber ;  the  variations  of  air  pressure  due  to  the 
sound  wave  are  communicated  through  the  speaking  tube 
t  to  one  side  of  the  partition,  while  the  gas  supply  for  the 
burning  jet  j  is  on  the  other  side;  the  movements  of  the 
diaphragm  produce  changes  in  the  pressure  of  the  gas 
which  cause  the  height  of  the  flame  to  vary  accordingly. 


er 


BUB 


ar 


V 


Fig.  58.    Manometric  flame  records  of  speech  by  Nichols  and  Merritt. 

The  vibrations  of  the  flame  may  be  observed  in  a  revolving 
mirror  m. 

A  method  has  been  devised  by  Professors  Nichols  and 
Merritt  in  which,  by  the  use  of  acetylene  gas,  the 
flame  may  be  photographed ;  they  have  obtained 
valuable  and  interesting  results  in  the  study  of  the 
vowels  and  spoken  words ;  Fig.  58  shows  a  portion  of  a 
flame  record  of  the  vibrations  from  the  spoken  words 
"preposterous  and  Raritan  River.  The  top  line  is  the  syl- 
lable pre,  the  second  hne  the  syllables  ter-ous;  the  next 

74 


RECORDING  AND  PHOTOGRAPHING  SOUND  WAVES 


two  lines  show  parts  of 
rar-i-t,  w^hile  the  bottom 
Hne  represents  ri-v. 

The  method  of  recording 
sound  vibrations  by  photo- 
graphing the  acetylene 
flame  has  been  still  further 
developed  by  Professor  J. 
G.  Brown,  whereby  an  out- 
line of  the  wave  form  is  ob- 
tained.^^  Fig.  59  shows  rec- 
ords made  by  this  method, 
the  boundary  between  the 
light  and  dark  portions 
being  the  wave  form. 


Fig.  60.    Duddell's  o.scillograph. 


Fig.  59.    Vibrating  flame  records  of  sounds 
by  Brown. 


The  Oscillograph 

The  telephone  invented  by 
Bell  in  1876,  as  well  as  the  mi- 
crophone transmitter  of  Hughes 
(1878),  generates  electromagnetic 
waves  from  the  sound  waves  im- 
pressed upon  the  diaphragm  of 
the  transmitter.  These  waves 
may  be  received  by  the  oscillo- 
graph, a  specialized  type  of  gal- 
vanometer. Fig.  60,  developed 
by  Blondel  (1893)  and  by  Dud- 
dell.  The  electric  waves  set  a 
minute  ixdrror  into  correspond- 
ing vibrations  which  may  be  re- 
corded'  [photographically.  The 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


telephone  receiver  proves  that  these  vibrations  correspond 
to  the  sound  waves  sufficiently  to  make  speech  intelligible, 
but  it  is  known  that  mechanical  and  electromagnetic  factors 
produce  appreciable  alterations  of  the  wave  forms. 

The  method  is  being  continually  developed  and  improved 
and  is  of  much  value,  especially  in  telephone  research.  Fig. 


0 


a 

Fig.' 61.    Records  of  vowels  obtained  with  the  oscillograph. 


61  shows  a  telephone-oscillograph  record  of  the  vowel  sounds 
a  and  o.^^ 

The  Phonograph 

The  phonograph,  invented  by  Edison  in  1877,  originally 
recorded  the  movements  of  a  diaphragm  by  indentations 

in  a  sheet  of  tin- 
foil supported  over 
a  spiral  groove  in 
a  metal  cylinder, 
Fig.  62.  In  later 
machines  the 
movements  of  the 
diaphragm  are  recorded  by  minute  cuttings  on  the  surface 
of  a  wax  cylinder  or  disk. 

76 


Fig.  62.    Phonograph,  early  form. 


RECORDING  AND  PHOTOGRAPHING  SOUND  WAVES 


A  modification  of  the  phonograph  was  invented  by  Bell 
and  Tainter  and  called  the  graphophone  :  Berliner  intro- 
duced the  method  of  etching  the  original  record  on  a  zinc 
disk,  producing  the 


gramophone. 

Hermann  in 
1890,  and  Bevier 
in  1900,  each  made 
photographic  cop- 
ies of  phonograph 
records  on  an  en- 
larged scale.  A  del- 
icate tracing  point 
carrying  a  mirror 
was  so  mounted 
that,  as  it  passed  slowly  over  the  record,  a  beam  of  light 
reflected  from  the  mirror  fell  upon  a  mo\4ng  photographic 
paper  or  film  and  registered  the  wave  form.  Fig.  63  shows 
records  of  vowel  sounds  obtained  by  Bevier. 


Fig.  63. 


256 


Vowel  curves  enlarged  from  a  phonographic 
record. 


A/'/^K  for  ftotet^ng 
Tutt 

Fig.  64.    Scripture's  apparatus  for  tracing  talking-machine  records. 

B}^  means  of  a  tracing  apparatus,  a  top  view  of  which  is 
shown  in  Fig.  64,  Scripture  has  copied  talking  machine 
records  enlarged  300  times  laterally  and  about  5  times  in 
length.^^    A  disk  record  is  rotated  very  slowly,  one  turn 

77 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


in  5  hours,  while  a  tracing  point  rides  smoothly  in  the  groove ; 
by  means  of  a  system  of  delicate  compound  levers,  the  lateral 
movements  of  the  tracer  are  registered  on  a  moving  strip 
of  smoked  paper.  The  whole  apparatus  is  operated  by  an 
electric  motor,  and  when  started,  may  be  left  to  continue 
the  tracing  to  the  end,  which  operation,  with  the  mag- 
nification em- 
ployed, is  prac- 
ticable for  a  few 
turns  only  of  the 
disk.  Fig.  65 
shows  such  a 
tracing  from  a 
record  of  orches- 
tral music,  which 
as  here  repro- 
duced is  magni- 
fied about  150 
times  laterally 
and  2 1  times  in 
length.  These 
copies  are  prob- 
ably the  best  that  have  been  obtained  from  phonographic 
records.  • 

Both  the  process  of  making  the  original  record  in  wax 
and  the  subsequent  enlarging  introduce  imperfections  into 
the  curves ;  nevertheless  these  methods  have  been  of  great 
value  in  many  researches  in  acoustics. 


Fig.  65. 


A  tracing,  by  Scripture,  of  a  record  of  or- 
chestral music. 


The  Phonodeik 

For  the  investigation  of  certain  tone  qualities  referred 
to  in  Lecture  VI,  the  author  required  records  of  sound  waves 


78 


RECORDING  AND  PHOTOGRAPHING  SOUND  WAVES 


showing  greater  detail  than  had  heretofore  been  obtained. 
The  result  of  many  experiments  was  the  development  of 
an  instrument  which  photographically  records  sound  waves, 
and  which  in  a  modified  form  may  be  used  to  project  such 
waves  on  a  screen  for  public  demonstration ;  this  instru- 
ment) has  been  named  the  ''Phonodeik,"  meaning  to  show 
or  exhibit  sound.^- 

The  sensitive  receiver  of  the  phonodeik  is  a  diaphragm, 
d,  Fig.  66,  of  thin  glass  placed  at  the  end  of  a  resonator 
horn  h\  behind  the  diaphragm  is  a  minute  steel  spindle 
mounted  in  jeweled  bearings,  to  which  is  attached  a  tiny 
mirror  m ;  one  part  of  the  spindle  is  fashioned  into  a  small 


Fig.  66.    Principle  of  the  phonodeik. 

pulley ;  a  few  silk  fibers,  or  a  platinum  wire  0.0005  inch 
in  diameter,  is  attached  to  the  center  of  the  diaphragm  and 
being  wrapped  once  around  the  pulley  is  fastened  to  a  spring 
tension  piece;  light  from  a  pinhole  I  is  focused  by  a  lens 
and  reflected  by  the  mirror  to  a  moving  film  /  in  a  special 
camera.  If  the  diaphragm  moves  under  the  action  of  a 
sound  wave,  the  mirror  is  rotated  by  an  amount  propor- 
tional to  the  motion,  and  the  spot  of  light  traces  the  record 
of  the  sound  wave  on  the  film,  in  the  manner  of  the  pendulum 
shown  in  Fig.  11,  page  12. 

In  the  instrument  made  for  photography.  Fig.  67,  the 
usual  displacement  of  the  diaphragm  for  sounds  of  ordinary 
loudness  is  about  half  a  thousandth  of  an  inch,  resulting  in 

79 


thp:  science  of  musical  sounds 


RECORDING  AND  PHOTOGRAPHING  SOUND  WAVES 


an  extreme  motion  of  one  thousandth  of  an  inch,  which  is 
magnified  2500  times  on  the  photograph  by  the  mirror 
and  fight  ray,  giving  a  record  2^  inches  wide ;  the  film 
commonly  employed  is  5  inches  wide,  and  the  record  is 
sometimes  wider  than  this.  The  extreme  movement  of 
the  diaphragm  of  a  thousandth  of  an  inch  must  include 
all  the  small  variations  of  motion  corresponding  to  the 
fine  details  of  wave  form  which  represent  musical  quality. 
Many  of  the  smaller  kinks  shown  in  the  photographs,  such 
as  Figs.  110  and  169,  are  produced  by  component  motions 
of  the  diaphragm  of  less  than  one  hundred-thousandth  of 
an  inch  ;  the  phonodeik  must  faithfully  reproduce  not  only 
the  larger  and  slower  components,  but  also  these  minute 
vibrations  which  have  a  frequency  of  perhaps  several 
thousand  per  second. 

The  fulfillment  of  these  requirements  necessitates  unusual 
mechanical  delicacy ;  the  glass  diaphragm  is  0.003  inch 
thick,  and  is  held  lightly  between  soft  rubber  rings,  which 
must  make  an  air-tight  joint  with  the  sound  box ;  the  steel 
staff  is  designed  to  have  a  minimum  of  inertia,  its  mass  is 
less  than  0.002  gram  (less  than  ^\  grain)  ;  the  small  mirror, 
about  1  millimeter  (0.04  inch)  square,  is  held  in  the  axis 
of  rotation ;  the  pivots  must  fit  the  jeweled  bearings  more 
accurately  than  those  of  a  watch ;  there  must  be  no  lost 
motion,  as  this  would  produce  kinks  in  the  wave,  which  when 
magnified  would  be  perceptible  in  the  photograph ;  there 
must  be  no  friction  in  the  bearings. 

The  phonodeik  responds  to  10,000  complete  vibrations 
(20,000  movements)  per  second,  though  in  the  analytical 
work  so  far  undertaken  it  has  not  been  found  necessary  to 
investigate  frequencies  above  5000. 

The  author  mshes  to  record  that  the  success  of  the  phono- 
G  81 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


deik  in  meeting  these  requirements  is  due  to  the  friendly 
interest  and  exceptional  skill  of  Mr.  L.  N.  Cobb,  who  con- 
structed the  steel  staff  in  accordance  with  designs  which 
seemed  almost  impracticable  and  then  mounted  it  in  perfect 
jeweled  bearings. 

The  camera  is  arranged  for  moving  films  of  5  inches  in 
width  and  of  lengths  to  100  feet ;  there  are  three  separate 
revolving  drums  having  circumferences  of  1,  2,  and  5  feet 
respectively ;  there  is  also  a  pair  of  drums,  each  holding  100 
feet  of  film,  arranged  for  winding  the  film  from  one  to  the 
other  during  exposure.  The  single  drums  are  turned  by 
an  electric  motor,  with  film  speeds  varying  from  1  to  50 
feet  per  second. 

A  rheostat  for  controlling  the  speed  of  the  motor  is  placed 
where  it  can  be  reached  by  the  experimenter  when  he  stands 
near  the  horn,  and  there  is  visible  a  tachometer  which  indi- 
cates the  film  speed.  For  general  display  pictures  a  speed 
of  5  feet  per  second  is  convenient,  while  for  records  to  be 
analyzed  40  feet  per  second  is  suitable ;  for  the  latter  pur- 
pose a  short  record  1  or  2  feet  long,  made  in  4V  or  2V  ^ 
second,  is  sufficient. 

The  camera  is  provided  with  several  shutters  of  various 
types  for  hand,  foot,  and  automatic  electric  release,  and  for 
any  desired  time  of  exposure ;  and  a  commutator  on  the 
revolving  drum  may  be  used  to  open  and  close  the  shutter 
at  desired  points  in  its  revolution. 

Besides  the  record  of  the  wave  there  are  photographed 
on  the  film  simultaneously  a  zero  line  to  give  the  axis  of 
the  curve  for  analysis,  and  time  signals  from  a  stroboscopic 
fork,  y^o^  second  apart,  to  enable  the  exact  determination 
of  pitch  from  measurements  of  the  film.  The  axis  and 
time  signals  are  shown  in  Fig.  96  and  in  many  others ; 

82 


RECORDING  AND  PHOTOGRAPHING  SOUND  WAVES 


when  the  photograph  is  intended  for  display  only,  these 
records  are  sometimes  omitted. 

For  visual  observations  the  camera  is  provided  with  a 
horizontal  revolving  mirror  which  reflects  the  vibrating 
light  spot  upward  on  a  ground  glass  in  the  form  of  a  wave ; 
an  inclined  stationary  mirror  above  the  ground  glass  makes 
the  wave  visible  to  the  experimenter  while  the  sound  is 
produced.  The  speed  of  the  revolving  mirror  and  the 
dimensions  in  general  are  so  proportioned  that  the  wave 
appears  on  the  ground  glass  in  the  same  size  and  position 
as  when  photographically  recorded.  The  speed  of  the  motor 
may  be  adjusted  till  the  wave  appears  satisfactory  and  the 
film  speed  will  be  automatically  varied  to  correspond ;  the 
sound  is  altered  in  loudness  or  quality  as  desired ;  when  a 
suitable  wave  appears  on  the  ground  glass,  the  closing  of  an 
electric  key  or  the  pressure  of  the  foot  on  a  floor  trigger 
makes  the  photographic  exposure.  The  photographs  are 
all  taken  under  such  conditions  that  the  film  moves  from 
right  to  left,  giving  the  time  scale  in  a  positive  direction, 
and  that  a  positive  ordinate  of  the  curve  corresponds  to 
the  compression  part  of  the  air  wave. 

A  sound-recording  instrument  might  best  be  used  out  of 
doors,  on  the  roof  of  a  building  for  instance,  to  avoid  con- 
fusion of  the  records  by  reflection  from  the  walls ;  since  it 
is  not  convenient  to  work  in  such  a  place,  the  disturbing 
factors  of  the  laboratory  room  are  minimized  by  various 
precautions,  such  as  padding  the  walls  with  thick  felt  ; 
Fig.  68  shows  the  room  in  which  the  photographs  are  made ; 
the  phonodeik  with  the  receiving  horn  stands  on  a  pier, 
while  the  light  and  moving-film  camera  are  behind  the 
screen.  The  tuning  fork  which  flashes  the  time  signal  is 
shown  at  the  right. 

83 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


RECORDING  AND  PHOTOGRAPHING  SOUND  WA\'ES 


The  Demoxstratiox  Phoxodeik 

The  vibrator  of  the  phonodeik  employed  in  research  is 
very  niinute  and  dehcate,  and  its  small  mirror  reflects  too 
little  hght  to  make  the  waves  visible  to  a  large  audience. 
For  purposes  of  demonstration,  a  phonodeik  has  been  espe- 
cially constructed,  Fig.  69,  which  will  clearly  exhibit  the 
principal  features  of  ''living"  sound  waves.^^  The  sound 
from  a  voice  or  an  instrument  is  produced  in  front  of  the 
horn ;  the  movements  of  the  diaphragm  with  its  \Tibrating 
mirror  cause  a  vertical  line  of  light  which,  falhng  upon  a 
motor-driven  revohdng  mirror,  is  thrown  to  the  screen  in 
the  form  of  a  long  wave  ;  the  movements  of  the  diaphragm 
are  magnified  40,000  times  or  more,  producing  a  wave  which 
may  be  10  feet  wide  and  40  feet  long. 

With  this  phonodeik  a  number  of  experiments  may  be 
made  in  further  explanation  of  the  principles  of  simple 
harmonic  motion  and  wave  forms.  \Mien  the  revoh-ing 
mirror  is  kept  stationary,  the  spot  of  light  on  the  screen 
moves  in  a  vertical  line  as  the  diaphragm  ^ib^ates ;  though 
these  movements  are  superposed,  their  extreme  complexity 
is  shown  since  the  turning  points  are  made  evident  by 
bright  spots  of  hght.  If  the  mirror  is  slowly  turned  by 
hand,  the  production  of  the  harmonic  curve  by  the  combina- 
tion of  \dbratory  and  translatory  motions  is  demonstrated. 
With  a  tuning  fork  the  simplicity  of  the  sine  curve  is  exhib- 
ited ;  with  two  tuning  forks  the  combination  of  sine  curves 
is  sho^^Tl ;  the  imperfect  tuning  of  two  forks  is  demon- 
strated by  a  slowly  changing  wave  form  ;  the  relations  of 
loudness  to  amplitude  and  of  pitch  to  wave  length  may  be 
illustrated. 

The  projection  phonodeik  is  especially  suitable  for  exhibit- 
So 


THE  S(  IEX(  E  OF  MUSICAL  SOUNDS 


RECORDING  AND  PHOTOGRAPHING  SOUND  WAVES 


ing  the  characteristics  of  sounds  from  various  sources ;  as 
seen  on  the  screen  the  sound  waves  are  constantly  in  motion, 
changing  shape  and  size  with  the  sUghtest  alteration  in 
frequency,  loudness,  or  quality  of  the  source. 

(As  delivered  orally,  this  Lecture  was  illustrated  with 
many  photographs  of  sound  waves  and  also  by  the  pro- 
jection of  the  sound  waves  from  various  sources  upon  the 
screen.  The  greater  number  of  the  photographs  so  used  are 
reproduced  in  various  parts  of  this  book,  while  the  charac- 
teristics of  the  sources  of  sound  are  described  in  Lecture  VI.) 

Determination  of  Pitch  with  the  Phonodeik 

The  photographs  obtained  with  the  phonodeik  permit 
a  very  convenient  and  accurate  determination  of  pitch ; 
the  time  signals  are  given  by  a  standard  tuning  fork,  record- 
ing one  hundred  flashes  per  second ;  it  is  only  necessary  to 
compare  the  wave  length  and  the  time  intervals  to  obtain 
the  frequency.  Various  photographs,  ks  Fig.  96,  show  the 
time  signals. 

A  standard  clock  with  a  break-circuit  attachment  may 
be  made  to  record  signals  simultaneously  with  the  sound 
waves ;  by  counting  and  measuring,  the  number  of  waves 
per  second  may  be  determined  with  precision.  When  two 
sounds  are  being  compared  by  the  method  of  beats,  the 
exact  number  (including  fractions)  of  beats  per  second  may 
be  determined  by  photographing  the  beats  together  with 
the  time  signals. 

The  phonodeik  permits  accurate  tuning  of  all  the  harmonic 
ratios  ;  if  the  spot  of  light  is  observed  without  the  revolving 
mirror,  its  movements  take  place  in  a  straight  line ;  two 
tones  sounding  simultaneously  give  a  composite  wave  form, 
the  turning  points  of  which  are  visible  as  circles  of  extra 

87 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


brightness  on  the  hne,  hke  beads  on  a  string.  When  the 
ratio  of  the  component  tones  is  inexact,  there  is  a  constant 
change  of  wave  form  which  causes  the  beads  to  creep  along 
the  hne ;  when  the  ratio  is  exact,  the  wave  form  is  constant 
and  the  beads  are  stationary,  signifying  perfect  tuning. 

Photographs  of  Compression  Waves 

The  methods  for  recording  sound  so  far  described  show 
the  movements  of  a  diaphragm  produced  by  the  varying 
air  pressure  of  the  sound  wave.  Sound  waves  consist  of 
alternate  condensations  and  rarefactions  which  are  prop- 
agated through  space  with  a  velocity  of  1132  feet  per 
second ;  for  the  tone  middle  C  the  distance  from  one  com- 
pression to  the  next  is  about  four  feet.  It  would  be  very 
useful  indeed  if  photographs  could  be  obtained  of  ordinary 
sound  waves  in  air,  but  no  practicable  means  has  yet  been 
devised  for  photographing  waves  of  this  size. 

A  method  due  to  Toepler  has  been  successively  developed 
by  Mach,  Wood,  Foley  and  Souder,  and  Sabine,  by  which 
instantaneous  photographs  can  be  obtained  of  the  snapping 
sound  of  an  electric  spark  from  a  Ley  den  jar.  This  sound 
consists  of  a  single  wave  containing  one  condensation  and 
one  rarefaction,  the  wave  length  may  be  inch  or  less,  and 
the  sound  is  relatively  a  loud  one,  that  is,  the  change  in 
density  is  considerable.  If  while  such  a  sound  wave  is 
passing  over  a  photographic  plate  in  the  dark,  the  wave  is 
instantaneously  illuminated  by  a  single  distant  electric 
spark,  the  light  from  the  spark  will  be  refracted  by  the 
sound  wave  which  will  then  act  as  a  lens  and  register  itself 
on  the  plate.  There  must  be  one  miniature  flash  of  lightning 
to  make  the  sound,  a  sort  of  minute  clap  of  thunder,  and  a 
second  distant  flash  a  small  fraction  of  a  second  later  to 

88 


RECORDING  AND  PHOTOGRAPHING  SOUND  WAVES 


illuminate  the  thunder  wave  as  it  passes  outwards.  This 
method  is  not  suitable  for  recording  the  pecuharities  of 
ordinary  sounds  due  to  various  tone  qualities,  but  it  is 
very  useful  in  studying  some  features  of  wave  propagation. 

The  most  beautiful  photographs  of  this  kind  have  been 
recently  obtained  by  Professor  Sabine  and  applied  by  him 
to  the  practical  problem  of  auditorium  acoustics,  to  which 


Fig.  70,    Cross-sectional  model  of  a  theater,  with  the  photograph  of  a  sound  wave 
entering  the  auditorium. 

reference  was  made  in  Lecture  II.  A  small  cross-sectional 
model  of  the  auditorium  is  prepared,  as  shown  in  Fig.  70, 
and  the  photographic  plate  is  placed  behind  it ;  the  sound 
is  produced  on  the  stage,  at  x,  and  the  resulting  wave  pro- 
ceeds on  its  journey  into  the  auditorium,  moving  at  the 
rate  of  1132  feet  per  second. 

The  wave  length  in  the  experiment  is  about  2V  inch,  which 
is  equivalent  to  a  wave  length  of  two  feet  in  the  actual  audi- 

89 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Fig.  71.    Position  of  a  sound  wave  in  a  theater  too  second  after 
its  production  on  the  stage. 


Fig.  72.    Echoes  in  a  theater  developed  from  a  single  sound 
impulse  in  jq\  second. 


90 


RECORDING  AND  rilOTOGRAPHING  SOUND  WAVES 


torium,  corresponding  to  the  musical  tone  one  octave  above 
middle  C.  The  wave  in  the  real  auditorium  will  have 
reached  the  position  shown  in  the  figure  in  about  y|o  second. 
Various  reflected  waves  or  echoes  are  beginning  to  appear : 
ai  is  produced  by  the  screen  of  the  orchestra  pit,  ao  is  from 
the  main  floor,  and  is  from  the  orchestra  pit  floor.  Fig. 
71  shows  the  waves  about  yf^  second  later,  just  before  the 
main  wave  reaches  the  balcony,  and  Fig.  72  shows  the 
waves  second  after  the  production  of  the  original  sound, 
when  the  main  wave  has  reached  the  back  of  the  gallery. 
The  large  number  of  echo  waves  which  seem  to  come  from 
many  directions  are  actually  generated  by  the  one  original 
impulse.  The  multiple  echoes  continue  to  develop  with 
increasing  confusion,  until  the  sound  is  diffused  throughout 
the  auditorium,  producing  the  condition  called  reverbera- 
tion. 


91 


LECTURE  IV 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 

Harmonic  Analysis 

Curves  and  wave  forms  such  as  those  obtained  with  the 
phonodeik  are  representative  not  only  of  sound,  but  of 
many  other  physical  phenomena,  and  their  study  is  of 
general  importance  in  science.  While  inspection  and  simple 
measurement  will  often  give  some  information  concerning 
these  curves,  as  will  be  explained  later,  they  are  in  general 
too  complicated  for  interpretation  in  their  original  forms, 
and  several  methods  of  analysis  have  been  developed  which 
greatly  assist  in  our  understanding  of  them. 

In  the  wave  method  of  analysis,  often  used  in  optics,  the 
attention  is  directed  to  the  speed  and  direction  of  propaga- 
tion of  the  waves  in  the  medium  and  to  their  combined 
effects ;  in  the  harmonic  method  consideration  is  given 
primarily  to  the  vibratory  character  of  the  movements  of 
the  medium,  these  vibrations  being  regarded  as  compounded 
of  a  series  of  motions,  which  may  be  infinite  in  number, 
but  each  of  which  is  of  a  simple  definite  type. 

For  the  investigation  of  the  complex  curves  of  the  sounds 
of  music  and  speech,  the  harmonic  method  of  analysis  is  the 
most  suitable  and  convenient ;  it  is  based  upon  the  im- 
portant mathematical  principle  known  as  Fourier's  Theorem, 
the  statement  and  proof  of  which  was  first  published  in 
Paris,  in  1822,  by  Baron  J.  B.  J.  Fourier.^^    For  the  present 

92 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


purpose  Fourier's  theorem  may  be  stated  as  follows :  If 
any  curve  be  given,  having  a  wave  length  I,  the  same  curve 
can  always  be  reproduced  and  in  one  particular  way  only, 
by  compounding  simple  harmonic  curves  of  suitable  ampli- 
tudes and  phases,  in  general  infinite  in  number,  having  the 
same  axis,  and  having  wave  lengths  of  /,  ^l,  and  succes- 
sive aliquot  parts  of  / ;  the  given  curve  may  have  any  arbi- 
trary form  whatever,  including  any  number  of  straight 
portions,  provided  that  the  ordinate  of  the  curve  is  always 
finite  and  that  the  projection  on  the  axis  of  a  point  describ- 
ing the  curve  moves  always  in  the  same  direction.  Alany 
of  the  curves  studied  by  this  method  can  be  exactly  repro- 
duced by  compounding  a  limited  number  of  the  simple 
curves ;  for  sound  waves  the  number  of  components  re- 
quired is  often  more  than  ten,  and  rarely  as  many  as  thirty ; 
in  some  arbitrary^  mathematical  curves,  a  finite  number  of 
components  gives  only  a  more  or  less  approximate  repre- 
sentation, while  an  exact  reproduction  requires  the  infinite 
series  of  components. 

Fourier's  theorem  may  be  stated  in  mathematical  form 
in  the  Fourier  Equation  as  follows  : 

y  =  -l  ydx-\-\     °  I 
-|  ycos  dxcos—  <r  jj  ycos ——dx  cos— — ;  

In  this  equation  ij  is  the  ordinate  of  the  original  complex 
curve  at  any  specified  point  x  on  the  base  line,  and  /  is  the 
fundamental  wave  length.  The  principal  part  of  this  equa- 
tion is  a  trigonometric  series  of  sines  and  cosines  and  this 
(or  the  whole  equation)  is  often  referred  to  as  Fourier's 
Series. 

93 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


The  Fourier  equation  may  be  given  a  simpler  appear- 
ance by  writing  it  in  a  second  symbolic  form : 


The  term  is  a  constant  and  is  equal  to  the  distance 
between  the  chosen  base  line  and  the  true  axis  of  the  curve ; 
if  the  base  line  coincides  with  the  axis,  =  0,  and  this 
term  does  not  appear  in  the  equation  of  the  curve.  Since 
this  term  has  no  relation  to  the  shape  of  the  curve,  its  value 
is  not  required  in  sound  analysis  ;  the  method  for  evaluating 
it,  however,  is  described  on  page  107. 

The  other  terms  of  the  equation  occur  in  pairs,  as  ai  sin  d, 
hi  cos  6,  etc.,  and  each,  whether  a  sine  or  cosine  term,  rep- 
resents a  simple  harmonic  curve.  The  successive  simple 
curves  of  the  sine  series  evidently  repeat  themselves  with 
frequencies  of  1,  2,  3,  etc.,  that  is,  they  have  wave  lengths 
in  the  proportions  of  1,  ^,  J,  etc.,  and  the  same  is  true  of  the 
cosine  series. 

Each  of  the  coefficients  ai,  bi,  a2,  62,  etc.,  is  a  number  or 
factor  indicating  how  much  of  the  corresponding  simple 
harmonic  curve  enters  into  the  composite ;  that  is,  it  shows 
the  amplitude,  or  height,  of  the  simple  wave.  For  the  re- 
production of  a  given  curve  it  may  happen  that  certain  of 
the  simple  curves  are  not  required,  and  the  corresponding 
coefficients  then  have  the  value  zero  and  their  terms  do  not 
appear  in  the  Fourier  equation  of  the  curve. 

A  sine  and  a  cosine  curve  of  the  same  frequency  but 
with  independent  amplitudes,  such  as  the  pairs  of  curves 
in  the  Fourier  equation,  can  be  compounded  into  a  single 
sine  (or  cosine)  curve  of  like  frequency  which  starts  on  the 


ai  sin  ^  +  a2  sin  2  ^  +  ^3  sin  3  ^  -(-  .  .  . 


61  cos  ^  +  &2  cos  2  6     hs  cos  3  ^  +  .  .  . 


II 


94 


AXAIASIS  AND  SYNTHESIS  OF  HAR:M0NIC  CURVES 


axis  at  a  point  different  from  that  of  the  component  curves, 
and  which  has  an  ampUtude  dependent  upon  the  amphtudes 
of  the  components.  The  relation  of  the  starting  point  of 
the  new  curve  to  that  of  its  components  is  called  its  phase, 
as  is  explained  on  page  126.  This  principle  may  be  stated 
in  symbols  as  follows,  a  and  h  being  the  amplitudes  of  the 
given  curves,  and  .4  that  of  the  resultant,  and  P  the  phase 
of  the  new  curve  : 


when 
and 


a  sin  ^  +  6  cos  ^  =  A  sin  (d  +  P), 

b 
a 


tan  P 


If  the  amplitudes  a  and  b  are  made  the  base  and  altitude, 
respectively,  of  a  right  triangle,  Fig.  73,  then  the  hypothe- 
nuse  is  the  ampUtude 
A  of  the  resultant 
curve  and  the  angle 
which  the  hypothe- 
nuse  makes  with  the 
base  is  the  phase. 

If  each  pair  of  sine 
and  cosine  terms  of 
the  general  Fourier 
equation  is  reduced  in  this  manner,  and  if  the  origin  is  on 
the  axis  of  the  curve,  the  equation  may  be  put  into  the 
following  equivalent  form,  consisting  of  a  single  series  of 
sines : 

2/=Aisin((9+Pi)  +A2sin(2^ +P2)  +A3sin(3^ +P3)  +  .  .  .  Ill 

In  this  equation  A 1  is  the  amplitude  of  the  first  component 
(the  fundamental  tone)  and  Pi  is  its  phase;  while  A2 

95 


Fig.  73.    Amplitude  and  phase  relations  of  compo- 
nent and  resultant  simple  harmonic  motions. 


THE  SCIFACE  OF  MUSICAL  SOUNDS 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVP:S 


and  P2  determine  the  second  component  (first  overtone  or 
octave),  etc. 

Form  III  of  the  Fourier  equation  is  most  suitable  for 
representing  the  results  of  the  physical  analysis  of  a  sound, 
though  the  actual  numerical  analysis  is  obtained  in  the  first 
form,  I,  of  the  equation. 


The  process  of  analyzing  a  curve  consists  of  finding  the 
particular  numerical  values  of  the  coefficients  of  the  Fourier 
equation  so  that  it  will  represent  the  given  curve.  Fourier 
showed  how  this  may  be  done  by  calculation  (see  page  133), 
but  as  it  is  a  long  and  tedious  process,  requiring  perhaps 
several  days'  work  for  a  single  curve,  various  mechanical 
devices  have  been  constructed  to  lessen  the  labor. 

The  coefficients  of  the  various  terms,  the  quantities  in 
square  brackets  in  equation  I,  have  the  following  form,  n 
being  the  order  of  the  term  : 


These  are  represented  by  ai,  bi,  etc.,  in  equation  II,  and  are 
the  amplitudes  of  the  component  simple  harmonic  curves. 
Each  definite  integral  is  the  area  of  a  certain  auxiliary 
curve  on  the  base  I,  the  nature  of  which  need  not  be  de- 
scribed here  ;  this  area,  divided  by  I,  gives  the  mean  height 
of  the  auxiliary  curve,  which  is  then  multiplied  by  2,  giving 
the  amplitude  of  the  corresponding  component.  There  are 
various  area-integrating  machines,  known  in  their  simple 
forms  as  planimeters,  which  can  be  adapted  to  the  deter- 
mination of  the  areas  of  a  given  curve  under  such  conditions 
as  to  indicate  on  the  dials  the  numerical  values  of  the 


Mechanical  Harmonic  Analysis 


2 


2 


H 


97 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Fourier  coefficients ;  in  some  machines  the  dial  readings 
are  the  coefficients,  in  others  the  dial  readings  require 
further  slight  reduction.  Such  machines  are  called  har- 
monic analyzers.  ' 

Several  types  of  harmonic  analyzers  are  briefly  referred 
to  on  page  128.  The  analyzer  devised  by  Professor  Henrici, 
of  London,  in  1894,  based  on  the  rolling  sphere  integrator, 
is  perhaps  the  most  precise  and  convenient  yet  made.*^  An 
instrument  of  this  type  used  by  the  author  in  the  study 
of  sound  waves,  is  shown  in  Fig.  74,  and  its  operation  will 
be  described. 

The  curve  to  be  analyzed,  which  must  be  drawn  to  a 
specified  scale,  as  is  explained  later,  is  placed  underneath 
the  machine ;  the  handles  h  are  grasped  with  the  fingers, 
and  the  stylus  s  is  caused  to  trace  the  curve,  which  re- 
quires movements  in  two  directions.  The  machine  as  a 
whole  rests  on  rollers  which  permit  it  to  be  moved  to  and 
from  the  operator,  in  the  direction  of  the  amplitude  of  the 
curve,  and  the  stylus  is  attached  to  a  carriage  which  rolls 
along  a  transverse  track  t  in  the  direction  of  the  length  of 
the  curve. 

The  instrument  shown  has  five  integrators ;  each  sphere, 
made  of  glass,  rests  on  a  roller  so  that  when  the  curve  is 
traced,  the  sphere  is  rotated  on  a  horizontal  axis  by  an 
amount  proportional  to  the  amplitude  of  the  curve ;  two 
integrating  cylinders  with  dial  indexes  rest  against  each 
sphere  at  points  90°  apart,  Fig.  75,  and,  by  means  of  a  wire 
and  pulley  w  are  given  rotation  about  a  vertical  axis  pro- 
portional to  the  movement  along  the  axis  of  the  curve. 
While  each  sphere  rolls  only  in  amplitude,  the  cylinders 
sliding  around  the  sphere  take  up  components  of  the 
amplitude  motion  which  are  proportional  to  the  sine  and 

98 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


cosine  of  the  phase  change  respectively.  The  first  inte- 
grator turns  once  around  its  sphere  while  the  tracer  moves 
over  one  wave  length  of  the  fundamental  curve,  that  is, 
while  the  stylus  is  being  moved  the  length  of  the  track  t, 
the  next  integrator  turns  twice,  and  the  others  three,  four, 
and  five  times  in  the  same  interval.     In  this  manner  one 


Fig.  75.    The  rolling-sphere  integrator  of  the  harmonic  analyzer. 


tracing  gives  the  ten  coefficients,  five  sines  and  five  cosines, 
of  the  first  ten  terms  of  the  complete  Fourier  equation  of  the 
curve. 

In  the  Henrici  analyzer  the  sizes  of  the  various  parts 
are  so  proportioned  that  the  effects  of  the  constant  factors 
of  the  amplitude  terms  are  mechanically  incorporated  in  the 
dial  readings,  which  are,  without  reduction  (except  for  the 

99 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


factor  n,  mentioned  below),  the  actual  amplitudes  in  milli- 
meters of  the  components  of  the  curve  traced.  When  the 
stylus  has  been  moved  over  one  wave  length  of  the  funda- 
mental, it  must  have  moved  over  two  wave  lengths  of  the 
second  component,  three  of  the  third,  and  so  on;  then  the 
integrator  for  the  second  component  has  integrated  two 
waves,  and  the  dial  readings  are  twice  the  required  coeffi- 
cients ;  in  general,  the  readings  of  the  nth  integrator  are  n 
times  too  large,  they  are  na„,  and  In  the  study  of 

sound  waves  the  presence  of  the  factor  n  is  a  convenience, 
for  the  quantities  finally  desired  are  the  intensities  of  the 


Fig.  76.    Photograph  of  the  sound  from  a  violin. 


components,  and,  as  explained  on  paga  167,  the  loudness  of 
any  component  is  proportional  to 

By  changing  the  wire  to  the  smaller  pulleys  v  on  the 
integrators,  the  spheres  are  turned  six,  seven,  eight,  nine, 
and  ten  times  while  tracing  the  wave,  and  the  dials  indicate 
the  sine  and  cosine  coefficients  for  the  components  from 
six  to  ten. 

By  a  reconstruction  of  the  analyzer  (in  1910)  which  it 
was  necessary  to  carry  out  in  our  own  instrument  shop, 
the  operation  of  the  instrument  has  been  extended  from 
ten  to  thirty  components  with  precision,  six  tracings  being 
required  for  the  larger  number. 

The  analysis  of  the  sound  wave  from  the  tone  B4  =  995, 

100 


ANALYSIS  AND  S\^THESIS  OF  HARMONIC  CURVES 


played  on  the  E  string  of  a  violin,  will  be  considered.  This 
curve,  which  is  shown  in  Fig.  76,  is  comparatively  simple. 
When  the  curve  is  analyzed  with  the  machine,  the  opera- 
tion proceeds  in  accordance  with  the  method  shown  in  the 
Fourier  equation  I,  but  the  mechanical  integrators  give 
the  result  in  form  II,  and  the  actual  equation  read  from 
the  dials  is  as  follows : 

y  =  151  sin  ^  -  67  cos  <9  +  24  sin  2  ^  +  55  cos  2  6 
+  27  sin  3^+5  cos  3  6. 

The  analyzer,  which  has  five  integrators,  gives  at  the  same 
time  mth  the  above  the  coefficients  of  the  terms  involving 


Fig.  77.    Curve  of  a  violin  tone  and  its  sine  and  cosine  components. 

4  6  and  5  ^ ;  in  this  instance  the  latter  coefficients  are  very 
small,  and  for  simphcity  they  are  omitted.    In  other  words 

101 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


the  analysis  shows  this  curve  to  be  composed  of  three  com- 
ponents only,  each  of  which  is  represented  by  a  pair  of  sine 
and  cosine  terms. 

In  practical  work,  each  pair  of  sine  and  cosine  terms  is 
at  once  reduced  to  a  single  term,  but  for  the  sake  of  illus- 
tration the  graphic  interpretation  of  the  equation  in  its 
present  form  is  given  in  Fig.  77  ;  there  are  six  simple  curves, 
a  sine  and  a  cosine  curve  for  each  of  the  three  frequencies, 
all  starting  from  the  same  initial  line  ah;  the  sine  curves 
are  indicated  by  Si,  S2,  and  S3,  and  the  cosines  by  Ci,  C2, 
and  C3.  These  six  curves  added  together,  or  made  into  a 
composite,  will  accurately  reproduce  the  violin  curve  V. 

As  explained  on  page  95,  each  pair  of  these  curves  can 
be  reduced  to  a  single  equivalent  curve,  and  the  six  com- 
ponents thus  become  three.  The  reduction  for  the  first 
pair  of  terms  gives  the  equation : 


^\iy       are  similarly  reduced,  giving 

Fig.  78.    The  resultant  of  sine  and     the    CUrVCS    for   the    first  and 


Fourier  equation  for  the  violin  curve,  in  form  III,  is  : 

2/  =  165  sin  ((9  +  336°)  +  60  sin  (2  ^  +  66°)  +  27  sin  (3  ^  +  11°). 

This  is  the  form  of  equation  usually  desired  in  physical 
investigations ;    its  graphical  interpretation  is  shown  in 


151  sin  ^  -  67  cos  ^  =  165  sin  ((9  +  336°). 


a 


The  graphic  interpretation  of 
this  reduction  is  shown  in  Fig. 
78,  in  which  Vi  is  the  resultant 
of  Si  and  Ci  and  is  the  true 
representation  of  the  funda- 
mental of  the  violin  curve.  The 
second  and  third  pairs  of  curves 


cosine  curves. 


second  overtones,  and  the  final 


102 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


Fig.  79,  which  shows  the  original  viohn  curve  at  the  top, 
wdth  its  three  true  components,  representing  partial  tones, 
drawn  separately  to  show  the  amplitudes  and  phases  (start- 
ing points)  more  distinctly. 


Fig.  79.    Curve  of  a  violin  tone  and  its  three  harmonic  components. 

The  complete  analysis  and  synthesis  of  a  more  compli- 
cated curve  is  described  later  in  this  Lecture. 

Amplitude  and  Phase  Calculator 
The  reduction  of  the  double  Fourier  series  consisting  of 
sines  and  cosines  to  the  single  series  of  sines  wdth  differing 

103 


V 


THE  SCIENCE  OF  MUSICAL  SOUNDS 

phases  is  usually  carried  out  by  numerical  calculation,  as 
has  been  indicated.  The  need  for  a  more  expeditious 
method,  where  a  large  number  of  curves  are  being  analyzed, 
has  resulted  in  the  design  and  construction  in  our  own 
laboratory  of  a  machine,  Fig.  80,  which  accomplishes  the 
purpose  in  a  satisfactory  manner."^^    This  amplitude-and- 


FiG.  80.    Machine  for  calculating  amplitudes  and  phases  in  harmonic  analysis. 


phase  calculator  is  essentially  a  machine  for  solving  right 
triangles. 

The  machine  has  two  grooves  at  right  angles  to  each 
other,  provided  with  linear  graduations ;  in  the  grooves 
are  movable  shders  which  carry  the  graduated  hypothe- 
nuse  bar ;  one  end  of  the  hypothenuse  is  attached  to  a 
special  angle  measurer,  while  the  other  end  slides  through 

104 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


a  support  which  also  bears  an 
index  for  reading  the  length 
of  the  bar. 

The  pair  of  coefficients  a 
and  h  of  the  general  Fourier 
equation,  which  are  given  by 
the  analyzer,  are  set  off  as 
the  base  and  altitude,  respec- 
tively, of  the  triangle,  when 
the  length  of  the  hypothenuse 
is  the  amplitude  A  of  the  re- 
sultant. 

The  phase  of  the  resultant  curve,  which  is  determined  by 
the  equation, 

tan  P  =  -  , 


Fig.  81, 


Phase  angles  in  four  quad- 
rants. 


0  0/  18(0 

\+  \-  -  + 
0]  180  180  0 

1       \  a 

3160 1     1  I 

270 
^  9(1 

90 

Fig.  82. 


Scheme  for  measuring  phases  in  one  quad- 
rant. 


as  indicated  in  Fig.  81,  for  +  a,  + 
a  value  between  0°  and  90° ;  for 

105 


may  have  any 
value  from  0°  to 
360°,  since  a  and 
b  may  have  either 
the  positive  or  the 
negative  sign. 
For  the  same  nu- 
merical values  of 
a  and  b,  and  there- 
fore for  the  same 
value  of  A,  there 
may  be  four  dif- 
ferent values  of 
the  phase  angle ; 
b,  the  angle  will  have 
-a,  +  b,  the  angle 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


has  a  value  between  90°  and  180°  ;  for  -  a,  -  h,  it  lies 
between  180°  and  270°,  and  for  +  a,  -  b,  it  is  between 
270°  and  360°.  A  special  angle  measurer  with  four  gradua- 
tions of  a  quadrant  each  might  be  used  for  the  four  possible 
combinations  of  algebraic  signs  ;  Fig.  82  illustrates  a  scheme 
for  such  graduations. 


Fig.  83.    Finding  the  axis  of  a  curve. 


As  a  single  graduated  arc  may  be  provided  with  two  sets 
of  numbers,  one  on  either  side,  two  quadrant  graduations 
are  sufficient.  To  prevent  confusion  a  movable  cover  is 
provided  for  the  graduations ;  this  has  four  apertures  so 
shaped  that  any  one,  and  one  only,  of  the  four  sets  of  num- 
bers is  visible  at  one  time,  according  to  the  position  of  a 
spring  catch  attached  to  the  cover.    There  are  four  posi- 

106 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 

tions  for  this  catch  marked  with  the  possible  combinations 
of  algebraic  signs  of  a  and  b ;  when  the  cover  is  set,  one 
reads  the  true  phase  angle  without  any  reduction. 

Axis  OF  A  Curve 
If  the  axis  of  a  curve  is  unknown  and  is  required,  it  is 
necessary  to  determine  the  first  or  constant  term  of  the 


Fourier  equation,  forms  I  and  II ;  this  term  consists  of  the 
area  included  between  the  arbitrarily  assumed  base  line 
and  the  curve,  divided  by  the  base,  and  therefore  it  repre- 
sents the  mean  height  of  the  curve  from  the  base.  A  line 
drawn  through  the  curve  parallel  to  the  base  and  distant 
from  it  by  the  mean  height  of  the  curve,  will  be  the  true 
axis  of  the  curve,  since  that  part  of  the  area  between  the 
curve  and  this  axis  which  is  above  the  axis  must  be  equal 
to  that  which  is  below  the  axis. 


Fig.  84.    Finding  the  true  axis  of  a  curve  with  the  planimeter. 


107 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Let  it  be  required  to  find  the  axis  of  the  curve  shown  in 
Fig.  83.  When  no  base  Hne  is  given,  any  Une  parallel 
to  the  axis  may  be  used,  such  as  a  line  touching  the  crests 
or  troughs  of  two  waves,  AB,  or  any  other  line  through 
points  on  two  waves  which  are  in  the  same  phase.  The 
area  between  the  assumed  base  and  the  curve  is  measured 
with  a  planimeter  of  any  type ;  this  area  di\dded  by  the 
wave  length  is  the  distance  h  from  the  base  AB  to  the  true 
axis  A'B\  Fig.  84  shows  a  precision  planimeter  with  a 
rolling  sphere  integrator,  in  position  for  the  axis  determi- 
nation of  a  curve,  that  is,  for  finding  the  first  or  constant 
term  of  the  Fourier  equation. 

The  constant  term  gives  no  information  regarding  the 
nature  or  shape  of  the  curve,  it  merely  gives  its  position 
with  regard  to  the  base  line  incidentally  employed  in  draw- 
ing or  tracing  the  curve.  Ordinarily  this  term  is  not  re- 
quired in  sound  analysis. 

Enlarging  the  Curves 

For  use  with  the  Henrici  analyzer  it  is  necessary  that  the 
wave  length  of  the  curve  which  is  traoed  shall  be  such  that 
when  the  tracing  point  moves  along  its  guiding  tracks  a 
distance  equal  to  the  wave  length,  the  integrator  for  the 
first  term  shall  make  exactly  one  revolution  around  the 
rolling  sphere  ;  in  the  instrument  illustrated  the  wave  length 
must  be  400  millimeters,  about  16  inches. 

The  photographs  of  sound  waves  obtained  with  the 
phonodeik  have  wave  lengths  varying  from  25  to  100  milli- 
meters ;  these  waves  are  enlarged  with  the  apparatus  shown 
in  Fig.  85.  The  photographic  film  negative  of  the  wave  is 
placed  in  an  adjustable  holder  /,  on  an  optical-bench  pro- 
jection lantern,  the  curve  being  projected  on  a  movable 

108 


ANALYSIS  AND  SYNTHP^SIS  OF  HARMONK^  (  URM:S 


easel  e;  adjustments  are  made  until  the  projected  wave  is 
of  the  proper  size,  is  well  defined,  and  has  its  axis  horizontal ; 
the  curve  is  then  traced  with  a  pencil  on  a  sheet  of  paper. 
The  initial  point  is  chosen  merely  with  reference  to  con- 
venience in  determining  the  length  of  one  wave,  as  a,  Fig. 
96,  page  122,  where  the  curve  crosses  the  axis.  The  time  re- 
quired for  the  operation  of  enlarging  a  curve  is  less  than 
five  minutes. 

Thus  all  curves  as  analyzed  are  of  the  same  wave  length, 


Fig.  85.    Apparatus  for  enlarging  curves  by  projection. 


regardless  of  their  original  size  and  frequency,  and  as  they 
are  drawn  on  a  standard  sheet  of  paper,  19  by  24  inches, 
fihng  is  facilitated.  The  harmonic  synthesizer,  described 
later,  draws  curves  of  this  same  wave  length,  400  milli- 
meters, which  permits  a  direct  comparison  of  the  analyzed 
and  synthesized  curves. 

Some  of  the  simpler  analyzers  mentioned  later  may  be 
used  with  a  curve  of  any  size  such  as  the  original  photo- 
graph, but  the  results  read  from  the  machine  require  further 
reduction  for  each  individual  curve  and  component ;  as 

109 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


already  stated,  with  the  Henrici  analyzer,  the  machine 
readings  are  final,  requiring  no  reduction,  so  far  as  analysis 
is  concerned.  Where  many  curves  are  being  exhaustively 
studied  by  analysis  and  synthesis,  the  enlargement  to  stand- 
ard wave  length  is  not  a  disadvantage. 

Synthesis  of  Harmonic  Curves 

It  is  often  required  to  perform  the  converse  of  the  ana- 
lytical process  which  has  been  described,  that  is,  to  recom- 
bine  several  simple  curves  to  find  their  resultant  or  com- 
posite curve ;  this  is  harmonic  synthesis.  The  synthesis 
of  curves  can  be  accomplished  by  calculation  in  some  in- 
stances, and  always  by  graphic  methods  by  adding  the 
measured  ordinates  of  the  component  curves  and  plotting 
the  results ;  since  both  of  these  methods  are  laborious, 
machines  called  harmonic  synthesizers  have  been  designed 
to  faciUtate  the  work. 

A  harmonic  synthesizer  is  a  machine  which  will  generate 
separate  simple  harmonic  motions  of  various  specified  fre- 
quencies, amplitudes,  and  phases,  and  will  combine  these 
into  one  composite  motion  which  is  recorded  graphically. 

One  of  the  earliest  synthesizers  was  made  in  London 
about  1876  by  Lord  Kelvin  (see  page  129),  to  be  used  as  a 
tide-predicting  machine ;  it  is  based  upon  the  pin-and-slot 
device  described  in  Lecture  I.  A  cord  fixed  at  one  end, 
Fig.  86,  passes  around  several  pulleys  and  at  the  other  end 
is  attached  to  a  pencil,  which  makes  a  trace  on  a  moving 
chart.  The  pulley  a  is  attached  to  a  pin-and-slot  device, 
which  moves  up  and  down  with  a  simple  harmonic  motion ; 
the  cord  will  transmit  this  motion  to  the  pencil,  doubled  in 
amount ;  if  the  chart  moves  continuously,  the  trace  is  a 
simple  harmonic  curve  of  a  frequency  depending  upon  the 

no 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 

rapidity  with  which  the  crank-pin  is  rotated,  and  of  an  ampU- 
tude  depending  on  the  distance  of  the  pin  from  the  center 
of  its  crank ;  the  wave  length  of  the  curve  depends  upon 
the  speed  with  which  the  chart  moves.  If  another  pulley 
b  is  attached  to  a  second  pin-and-slot  device  rotating  twice 
as  fast  as  the  first,  it  will  give  the  pencil  a  simple  harmonic 
motion  of  twice  the  frequency  of  the  first. 
•  It  is  evident  from  the  manner  in  which  the  cord  passes 


around  the  system  of  pulleys  that  if  the  two  devices  operate 
simultaneously,  the  pencil  will  have  a  composite  motion 
which  is  the  sum  of  the  two  components,  and  the  trace  will 
be  the  synthetic  curve.  The  scheme  may  be  extended  to 
include  any  number  of  simple  harmonic  motions  of  any 
desired  frequencies,  phases,  and  amplitudes. 

Two  harmonic  synthesizers,  especially  for  the  study  of 
sound  waves,  have  been  designed  and  constructed  in  the 
laboratory  of  Case  School  of  Applied  Science,  one  having 


Fig.  86.    Apparatus  illustrating  the  method  of  harmonic  synthesis. 


Ill 


THE  SCIENCE  OF  INIUSICAL  SOUNDS 


112 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 

ten  components  and  the  other  thirty-two.  The  ten-com- 
ponent machine,  finished  in  1910,  was  soon  found  inade- 
quate for  the  study  of  musical  sounds  and  was  dismounted 
in  1914,  upon  the  completion  of  the  thirty-two-component 
synthesizer  shown  in  Fig.  87.  The  latter  machine  is  per- 
haps more  convenient  for  the  study  of  harmonic  curves  in 
general  than  any  other  which  has  been  constructed ;  it 
draws  curves  with  great  accuracy  and  on  a  very  large  scale ; 
the  drawing  board  is  24  by  34  inches  in  size,  but  by  shifting 
the  paper  and  pen  a  curve  of  almost  any  size  may  be  drawn. 
The  largest  single  component  curve  may  be  28  inches  wide 
and  have  a  wave  length  of  32  inches ;  the  highest  com- 
ponent may  be  4  inches  wide.  The  wave  length  commonly 
used  is  400  millimeters,  about  16  inches,  the  same  as  that 
used  with  the  analyzer,  but  larger  or  smaller  wave  lengths 
are  easily  arranged.  Thirty  of  the  elements  are  provided 
with  gears  giving  the  relative  frequencies  1,  2,  3  ...  30 ; 
the  other  two  elements  are  arranged  with  change-gears, 
like  a  lathe  head,  which  permits  their  easy  setting  for  higher 
or  lower  frequencies  or  for  inharmonic  frequencies.  The 
machine  can  be  quickly  set  to  give  the  frequencies  1,  2,  4,  6, 
and  all  even  terms  to  60;  or  for  the  series  1,  3,  6,  9,  and 
all  multiples  of  3  to  90.  The  mechanical  arrangements 
permit  the  amplitude  and  phase  of  any  component  to  be 
readily  set  to  any  value  ;  all  the  graduated  circles  and  scales 
are  on  the  upper  surface  of  the  machine  and  are  of  white 
celluloid.  There  are  special  scales  showing  the  phase  and 
amplitude  of  the  synthesized  curve.  All  the  motions  which 
affect  the  separate  components  as  they  are  being  com- 
pounded and  synthetically  drawn  are  provided  with  ball 
bearings,  to  eliminate  friction  and  lost  motion ;  the  motion 
is  so  accurately  transmitted  to  the  pen  that  a  wave  can  be 
I  113 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


clearly  drawn  in  which  the  amphtude  is  less  than  0.2  milli- 
meter (less  than  inch).  This  machine  is  described  in 
detail,  with  specimen  curves,  in  the  Journal  of  the  Franklin 
Institute.^^ 

A  ten-component  curve  can  be  synthesized  in  about  five 
minutes,  while  the  machine  may  be  set  for  thirty  com- 
ponents and  the  curve  drawn  in  twelve  minutes. 

In  the  study  of  sound  waves  the  synthesizer  is  chiefly 
used  to  verify  the  correctness  and  sufficiency  of  the  analyses. 
The  several  unit  devices  of  the  machine  having  been  set  to 
reproduce  the  separate  components  in  exact  sizes  and 
phases,  the  tracer  will  draw  the  resultant  curve.  If  this 
resultant  curve  is  exactly  hke  the  original  which  was  analyzed, 
the  analysis  is  correct  and  complete,  and  the  fact  is  recorded 
by  tracing  the  synthetic  curve  over  the  original  in  a  con- 
trasting color  of  ink.  Fig.  99,  page  127,  shows  the  synthetic 
reproduction  of  the  analysis  of  an  organ-pipe  curve  (Figs.  96 
and  98)  drawn  on  the  photograph  itself. 

The  synthesizer  is  also  useful  for  drawing  a  curve  corre- 
sponding to  the  average  of  several  photographed  curves, 
and  for  drawing  curves  of  any  assumed  composition,  as  in 
trial  analysis.  After  a  photographed  curve  has  been 
analyzed  and  the  components  have  been  corrected  for  in- 
strumental disturbances,  as  explained  in  Lecture  V,  it  is 
often  useful  to  draw  the  corrected  synthetic  curve,  as  is 
illustrated  in  Fig.  132,  page  173. 

The  synthesizer  is  useful  in  preparing  illustrations,  such 
as  many  of  those  required  in  these  lectures ;  it  would  be 
very  difficult  to  draw  the  curves  of  correct  form  by  any 
other  means. 

When  the  synthesizer  is  set  for  any  curve,  if  the  handle 
is  turned^till  the  phase  circle  for  the  first  component  reads 

114 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


0°,  the  circles  for  the  other  components  show,  without  cal- 
culation, the  relative  phases,  or,  as  sometimes  called,  the 
epochs,  of  the  several  components ;  the  tracing  point  will 
now  be  at  what  may  be  considered  the  initial  point  of  the 
wave,  though  in  general  this  will  not  be  where  the  curve 
crosses  the  axis.  These  relations  are  further  explained  in 
connection  with  the  analysis  of  the  curve  shown  in  Fig.  96, 
on  page  122. 

The  mathematician  finds  the  harmonic  synthesizer  useful 
for  the  investigation 
of  many  kinds  of 
curves ;  the  proper- 
ties of  periodic  func- 
tions and  the  COnver-  geometrical  form. 

gency  of  series  can  be  shown  graphically.  The  equation 
of  the  wave  form  made  up  of  two  straight  lines,  as  shown 
in  Fig.  88,  is  represented  by  the  infinite  series 

y  =  2  [sin  x  +  ^  sm2  x  +  |  sin  3  a;  +  J  sin  4  x  +  .  .  .], 

the  wave  length  being  equal  to  2  tt  =  6.28"^. 

The  manner  in  which  such  an  angular  geometrical  figure 
may  be  built  up  from  smooth  curves  is  shown  by  drawing 
curves  representing  different  numbers  of  terms  of  the 
series.  The  first  term  only,  y  =  2  sin  x,  is  represented  in 
Fig.  89,  a;  h,  c,  d,  e,  and  /  represent  the  curves  obtained 
when  two,  three,  four,  five,  and  ten  terms,  respectively,  are 
used.  Fig.  90  is  the  curve  obtained  when  thirty  terms  are 
included.  These  curves  are  graphic  illustrations  of  the 
convergence  of  this  series ;  the  more  terms  employed,  the 
closer  the  result  approximates  the  given  form ;  an  infinite 
number  of  terms  would  be  required  to  reproduce  the  figure 
exactly. 

115 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Fig.  89.    Forms  obtained  by  compounding  1,  2,  3,  4,  5,  and  10  terms  of  the  series 
y  =  2  [sin  x     \  sin  2  rc  +  |  sin  3  a;  +  .  .  .]. 

116 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


Fig.  91  is  a  curve  made  up  of  the  same  components  as 
enter  into  the  curve  shown  in  Fig.  90  ;  the  only  difference  is 


^^^^^^ 

Fig.  90.    Curve  obtained  by  cc 
sin 

aipounding  30  terms  of  the  ser 
2  X  +  -J  sin  3  X-  +  .  .  .]. 

es  ^  =  2  [sin  x  +  ^ 

that  the  phase  of  each  component  has  been  changed  by  90°  ; 
that  is,  the  sines  become  cosines. 

A  further  interesting  variation  is  obtained  by  using  the 


Fig.  91.  Curve  obtaiueJ  by  compounding  30  terms  of  the  series  y  =  2  [sin  (x + 
90°)  +  i  sin  (2  X  +  90°)  +  ^  sin  (3  x  +  90°)  +  .  .  .].  which  is  equivalent  to 
y  =  2  [cos  X  +  §  cos  2  x  +  ^  cos  3  x  +  .  .  .]. 

117 


THE  SCIENCE  OF  MUSICAL  SOUNDS 

odd-numbered  terms  only  of  the  first  series,  producing  the 
form  shown  in  Fig.  92. 

If  the  phases  of  the  alternate  terms  of  the  odd-term  series 


\/\/sy  — ..^s/N/y 

0 

77- 

\/\/w-~-  . — ✓w/vl 

277 

^Vw  -n^vaA 

Fig.  92.    Curve  obtained  by  compounding  15  terms  of  the  series  y  =  2  [sin  x  + 
^  sin  3  X  +  J  sin  o  x  +  .  .  .]. 

are  changed  by  180°,  the  curved  form  shown  in  Fig.  93  is 
obtained. 

The  arbitrary  nature  of  the  curves  that  may  be  studied 


Fig.  93.  Curve  obtained  by  compounding  15  terms  of  the  series  y  =  2  [sin  x  + 
i  sin  (3x+  180°)  +  i  sin  5  x  +  ^  sin  (7  x  +  180°)  +  ...],  or  y  =  2  [sin  x  - 
^  sin  3  X  +  I  sin  5  x  —  7  sin  7  x  +  .  .  .]. 


by  the  Fourier  method  is  further  illustrated  by  the  analysis 
and  synthesis  of  a  portrait  profile.  The  original  portrait 
is  shown  in  the  center  of  Fig.  94,  while  a  tracing  of  the 

118 


ANALYSIS  AND  SYNTHESIS  OF  HARIVIONIC  CURVES 


profile  is  given  at  the  left,  0.  The  curve  was  analyzed  to 
thirty  terms,  but  the  coefficients  of  the  terms  above  the 
eighteenth  were  negligibly  small.    The  equation  of  the 


0  s 

Fig.  94.    Reproduction  of  a  portrait  profile  by  harmonic  analysis  and  synthesis. 

curve  is  as  follows,  the  numerical  values  corresponding  to  a 
wave  length  of  400  : 

y  =     49.6  sin  (     6  +  302°)  +17.4  sin  (  2  6  +  298°) 

+  13.8  sin  (  3  ^  +  195°)  +  7.1  sin  (  4  (9  +  215°) 

+  4.5  sin  (  5  ^+  80°)  +  0.6  sin  (  6  (9+171°) 

+  2.7  sin  (  7  (9  +  34°)  +  0.6  sin  (  8  ^  +  242J 

+  1.6  sin  (  9(9  +  331°)  +  1.3  sin  (10  ^  +  208°) 

+  0.3  sin  (11  (9+  89°)  +  0.5  sin  (12  (9  +  229°) 

+  0.7  sin  (13  (9  +  103°)  +  0.3  sin  (14  0  +  305°) 

+  0.4  sin  (15  (9  +  169°)  +  0.5  sin  (16  (9  +  230°) 

+  0.5  sin  (17  e  +  207°)  +  0.4  sin  (18  6  +  64°). 


This  equation  was  set  up  on  the  synthesizer,  and  the  portrait, 
as  drawn  by  the  machine,  is  shown  at  the  right,  S,  Fig.  94. 

119 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


If  mentality,  beauty,  and  other  characteristics  can  be 
considered  as  represented  in  a  profile  portrait,  then  it  may 
be  said  that  they  are  also  expressed  in  the  equation  of  the 
profile. 

Since  the  profile  is  reproducible  by  compounding  a  num- 
ber of  simple  curves,  it  is  possible  to  compound  the  simple 
tones  represented  by  these  curves  in  such  a  way  that  the 
resulting  wave  motion  of  the  combined  sounds  shall  be  the 
periodic  repetition  of  the  profile.  Fig.  95  is  a  drawing  of 
such  a  wave.    The  reproduction  of  vowel  wave  forms,  shown 


Fig.  95.    Wave  form  obtained  by  repeating  a  portrait  profile. 


in  Fig.  181,  page  250,  is  a  similar  synthetic  experiment. 
In  this  sense  beauty  of  form  may  be  likened  to  beauty  of 
tone  color,  that  is,  to  the  beauty  of  a  certain  harmonious 
blending  of  sounds. 

The  Complete  Process  of  Harmonic  Analysis 

A  curve  having  been  provided,  such  as  the  photograph 
of  a  sound  wave,  an  electric  oscillogram,  a  diagram  of  baro- 
metric pressures,  or  a  chart  of  temperatures,  its  complete 
analysis  by  the  Fourier  harmonic  method  may  be  con- 
veniently carried  out  in  accordance  with  the  following 
scheme : 

(a)  The  curve  is  redrawn  to  the  standard  scale  required 
by  the  Henrici  analyzer,  so  that  the  wave  length  is  400 
millimeters.    Time  required  :  five  minutes. 

(6)  The  curve  is  traced  with  the  analyzer,  one  tracing 
giving  five  sine  and  five  cosine  coefficients  of  the  complete 
Fourier  equation  of  the  curve,  determining  five  components. 

120 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


By  changing  the  wire  attached  to  the  tracer  from  one  set 
of  pulleys  on  the  integrators  to  another  set,  a  second  tracing 
gives  five  more  pairs  of  coefficients,  determining  ten  com- 
ponents of  the  curve.  Continued  tracings  will  give  fifteen, 
twenty,  twenty-five,  and  thirty  components.  Time  re- 
quired :  for  each  tracing,  including  making  the  record,  five 
minutes  ;  for  ten  components,  ten  minutes. 

(c)  Each  pair  of  sine  and  cosine  terms  as  given  by  the 
analyzer  is  reduced  by  the  triangle  machine,  to  determine 
the  true  amplitude  of  the  corresponding  component,  together 
with  its  phase.  Time  required  :  for  ten  pairs  of  terms, 
including  making  the  record,  five  minutes. 

(d)  The  correctness  and  completeness  of  the  analysis  are 
verified  by  setting  the  synthesizer  for  the  values  of  the 
amplitudes  and  phases  of  the  several  components  and  then 
reproducing  the  original  curve.  Time  required :  for  ten 
components,  five  minutes ;  for  thirty  components,  twelve 
minutes. 

(e)  The  numerical  quantities  of  the  analysis  are  pre- 
served on  cards  suitable  for  filing;  the  synthetic  curve  is 
drawn,  superposed  on  the  original  curve,  forming  a  per- 
manent record  of  the  degree  of  approximation  secured. 
Time  required  :  included  in  the  time  given  for  the  operations 
(6),  (c),  id). 

(/)  The  synthesizer  may  be  used  to  draw  each  component 
separately,  in  its  true  amplitude  and  phase.  Time  re- 
quired :  for  ten  components,  twenty  minutes. 

(g)  The  true  axis  of  the  curve  may  be  determined  with 
the  planimeter.    Time  required  :  three  minutes. 

The  times  mentioned  for  the  several  operations  are  those 
required  when  a  number  of  curves  are  being  analyzed  in 
routine ;  if  a  single  curve  is  analyzed  by  itself,  a  longer  time 

121 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


will  be  consumed.  The  analysis  of  a  curve  as  ordinarily 
understood  involves  only  the  operations  (6)  and  (c),  re- 
quiring about  fifteen  minutes  for  ten  components. 

Card  forms,  Fig.  97,  5  by  8  inches  in  size,  have  been 
arranged  for  preserving  the  data  of  analysis  and  reduction, 
as  required  in  the  study  of  sound  waves.  One  card  con- 
tains the  data  for  ten  components.  The  cards  for  the  first 
ten  components,  n  =  1  to  n  =  10,  are  white  in  color ;  for 
a  larger  number  of  components,  cards  of  different  colors 
are  used,  buff  for  values  of  n  from  11  to  20,  and  salmon 
for  n  =  21  to  n  =  30 ;  blue  cards  are  used  for  additional 
information,  averages,  etc.  The  data  relating  to  any  com- 
ponent are  given  in  the  vertical  column  under  the  value  of 
n  corresponding  to  the  order  of  the  component. 


Example  of  Harmonic  Analysis 

As  a  further  illustration  of  harmonic  analysis,  let  it  be 
required  to  analyze  the  curve  of  an  organ-pipe  tone,  shown 


Fig.  96.    Photograph  of  the  sound  wave  from  an  organ  pipe. 


in  Fig.  96.  The  curve  is  traced  twice  with  the  analyzer, 
the  necessary  change  in  the  wire  being  made  between  the 
two  tracings ;  then  each  of  the  ten  pairs  of  sine  and  cosine 

122 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


terms  is  reduced  with  the  triangle  machine  for  finding  the 
resultant  amphtudes  and  phases.  The  actual  time  required 
for  the  complete  determination  of  the  ten  amplitudes  and 
ten  phases,  including  the  recording  of  the  results  on  the 
analysis  card,  was  thirteen  minutes.  (The  actual  analysis 
of  this  curve  was  extended  to  twenty  components,  but  it 
was  found  to  contain  only  twelve  components  of  appreciable 
size.) 

Fig.  97  is  a  reproduction  of  the  card  containing  the  com- 


CASE  SCHOOL  OF  APPLIED  SCIENCE                        DEPARTMENT  OF  PHY 

ANALYSIS  OF  SOUND-WAVES 
No.)t90l|Source  K),^,a,o^    'P^.  -."^UtAA*>^ 

SICS 

Datealp,vl5l2-3 

1-10       Tone          ^                ^      Abs.  N  2.&0 

Purpose  (V^i^a.^h.^^^, 

component,  n 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

na„ 

+  39.6 

+  /OJ.I 

-  49.1 

+  44.8 

-  II.S 

-  7.1 

nb„ 

t  ^-^.| 

7  87.Z 

-  7.f 

-  26.1 

-  lO.d) 

-  3.8 

-66.7 

-21.7 

nA„ 

/32.0 

;o9.4 

76.8 

45:0 

71.1 

k„.(  ) 

nA„k„ 

|nA„k„l' 

A„ 

/0.3 

6.4 

8.9 

4.3 

2.3 

P,, 

7c; 

33  7' 

330." 

354.° 

2.^0." 

252.' 

A„k„ 

Amplitude,?? 

Phase 

5-0.° 

310." 

zso: 

iszr 

4r 

Intensity.  ^ 

Remarks                                                                        Sum                 Analyzed  '^'^j^. 

Synthesized 

Fig.  97.    Card  form  for  the  record  of  the  analysis  of  a  sound  wave. 


plete  records  of  the  analysis  of  the  above  curve  (for  the 
first  ten  components).  The  first  two  fines,  nan  and  nbny 
are  the  coefficients  (each  multipfied  by  n,  the  order  of  the 
component)  of  the  sine  and  cosine  terms  of  the  Fourier 
equation,  form  II,  as  read  from  the  dials  of  the  analyzer. 
Each  pair  of  numbers  is  reduced  with  the  amplitude-and- 
phase  calculator,  giving  nA  „  and  Pn ;  each  of  the  multiple 
amplitudes,  nA„,  is  divided  by  the  corresponding  value  of 

123 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


n,  giving  An  and  Pn  are  the  coefficients  and  phases  of 
form  III  of  the  Fourier  equation. 

The  mathematical  equation  of  the  organ-pipe  curve 
(twelve  components)  is,  then,  as  follows,  the  wave  length 
being  equal  to  400 : 

y  =  Ao  +  96.5  sin  (  6+  76°)  +  66.0  sin  (  2  6  +  319°) 
+  36.5  sin  (  3  6>  +  337°)  +  19.2  sin  (  4  6*  +  354°) 
+  10.3  sin  (  5  6  +  330°)  +  8.4  sin  (  6  6  +  347°) 
+  6.4  sin  (  7  6  +  354°)  +  8.9  sin  (  8  ^  +  290°) 
+  4.3  sin  (  9  6  +  252°)  +  2.3  sin  (10  6  +  252°) 
+  2.2  sin  (11  ^  +  230°)  +  1.5  sin  (12  (9  +  211°) 

The  graphic  interpretation  of  this  equation  is  given  in 
Fig.  98.  The  equation  as  a  whole  is  represented  by  the 
original  curve  at  the  top ;  each  of  the  twelve  sine  terms 
corresponds  to  one  of  the  simple  curves  1,  2,  .  .  .  12.  The 
numerical  values  of  the  several  coefficients  (96.5,  66.0,  etc.) 
are  the  actual  amplitudes,  Ai,  A 2,  etc.,  of  the  component 
curves,  expressed  in  millimeters,  for  a  wave  length,  ah,  of 
400  millimeters.  If  the  curve  is  drawn  to  any  other  scale, 
the  coefficients  must  be  changed  in  the  same  proportion  as 
is  the  wave  length.  The  phases  of  the  several  components 
(76°,  319°,  etc.)  express  the  positions  of  the  curves,  length- 
wise, with  respect  to  the  initial  line,  ai. 

The  Une  photographed  as  the  axis  often  is  not  the  mathe- 
matical axis  of  the  curve.  The  true  axis  is  found,  as  de- 
scribed on  page  107,  with  the  planimeter.  The  curve  was 
traced  with  the  planimeter,  showing  that  the  area  of  that 
part  of  the  curve  which  lies  below  the  horizontal  line  exceeds 
that  above  by  2704  square  millimeters ;  this  quantity, 
divided  by  the  wave  length,  400,  gives  6.75  millimeters  as 
the  distance  of  the  true  axis  below  the  assumed  axis.  The 
true  axis  is  the  dotted  Une        in  Fig.  98 ;  if  the  curve  is 

124 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


11  r 

12  y- 


i        z  w 

Fig.  98.    An  organ-pipe  curve  and  its  harmonic  components. 


125 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


traced  with  respect  to  this  Hne,  the  areas  above  and  below 
the  Hne  will  be  found  equal.  The  distance  of  the  true  axis 
from  the  assumed  line,  —  6.75,  is  the  numerical  value  of 
the  coefficient       of  the  equation  of  the  curve. 

The  analysis  means  that  the  original  curve  is  equivalent 
to  the  simultaneous  sum,  or  composite,  of  the  several  com- 
ponent curves.  If  the  axes  of  the  twelve  components  all 
coincided  with  the  axis  a'b',  then  the  algebraic  sum  of  the 
twelve  ordinates  of  the  curves  at  any  point  x  (along  the 
line  yz)  would  be  equal  to  the  ordinate  xy  of  the  original 
curve.  The  ordinate  of  the  first  component  for  the  point 
a:  is  +  Xiyi ;  for  the  second  component  it  is  +  ^2^/2 ;  for  the 
third  it  is  zero,  Xz ;  for  the  fourth  it  is  —  X42/4,  etc. ;  the  sum 
is  positive  and  equal  to  xy.  For  the  point  u,  the  sum  of 
the  ordinates  of  the  components  on  the  line  uw  is  zero, 
that  is,  the  curve  crosses  the  axis  at  this  point.  This 
graphic  representation  of  the  analysis  of  a  curve  is  in  accord- 
ance with  the  principles  illustrated  in  the  models  of  three 
waves  shown  in  Lecture  II,  page  59.  If  the  separate  simple 
sounds  from  twelve  tuning  forks  (or  other  source)  produce 
motions  in  the  air  represented  by  the  twelve  component 
curves,  then  the  composite  tone  of  all  would  produce  a 
composite  motion  represented  by  the  original  curve. 

The  meaning  of  the  phases  (or  epochs)  of  the  several 
components  may  be  further  explained  by  reference  to  the 
figure.  The  starting  point  for  tracing  with  the  analyzer  is 
arbitrarily  selected ;  it  may  be,  for  instance,  the  point  a, 
Figs.  96  and  98,  where  the  photographed  curve  crosses  the 
photographed  axis,  which  may  or  may  not  be  the  true 
axis.  The  phases  obtained  by  analysis  then  give  the  rela- 
tions of  the  several  component  curves  to  the  assumed  initial 
line  ai.     In  the  example  here  shown,  the  phase  of  the  first 

.126 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


component  is  76°  ;  this  means  that  where  the  curve  crosses 
the  initial  hne,  the  first  component  has  already  progressed 
76°  (one  wave  length  equals  360°)  from  its  own  zero  point. 
The  zero  point  for  the  first  component  is  then  of  the 
wave  length  to  the  left  of  the  initial  line,  at  c  in  Fig.  98. 
The  phases  of  the  other  components  have  similar  interpre- 


FiG.  99.    Proof  of  the  analysis  of  a  curve  by  synthesis. 


tations,  each  being  measured  in  terms  of  its  own  wave 
length. 

It  is  sometimes  desirable  to  consider  the  beginning  of  a 
curve  as  the  point  where  the  phase  of  the  first  component 
is  zero ;  the  initial  line  would  then  be  at  the  point  d,  and 
the  wave  length  would  be  de.  This  point,  in  general,  is 
not  where  the  curve  crosses  the  axis  and  there  is  no  way  of 
determining  it  in  advance  of  analysis. 

The  phases  of  the  several  higher  components  at  the 

127 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


point  where  that  of  the  first  component  is  zero  are  con- 
venient for  the  comparison  of  phases ;  these  are  obtained 
by  subtracting  from  each  phase,  as  obtained  by  analysis, 
n  times  the  phase  of  the  first  component.  These  relative 
phases  are  determined  without  calculation  with  the  syn- 
thesizer as  explained  on  page  114,  and  they  are  recorded 
on  the  card  in  the  fine  labelled  ''Phase"  as  shown  in 
Fig.  97. 

The  verification  of  an  analysis  is  made  by  synthesis.  The 
equation  of  the  curve  is  set  up  on  the  synthesizer,  and  the 
curve  is  drawn  by  machine,  superposed  on  the  enlarged 
drawing  of  the  curve  which  was  used  with  the  analyzer. 
For  illustration  in  this  instance,  the  original  photograph 
has  been  enlarged  and  the  synthetic  curve  has  actually 
been  drawn  by  machine  on  the  photograph.  Fig.  99  is  a 
reproduction  of  the  original  and  synthetic  curves.  The 
likeness  is  sufficiently  close  for  general  purposes ;  a  more 
exact  reproduction  would  probably  require  the  inclusion  of 
a  very  large  number  of  higher  components,  all  of  which 
have  very  small  amplitudes. 


Fig.  100.    Kelvin's  tidal  harmonic  analyzer. 


Various  Types  of  Harmonic  Analyzers  and  Synthe- 
sizers 

The  general  methods  of  harmonic  analysis  and  synthesis 
which  have  been  described  in  detail  in  the  preceding  pages 

128 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


are  applicable  to  all  kinds  of  investigations  requiring  such 
treatment.  However,  variations  of  the  mechanical  devices 
are  sometimes  desired  to  obtain  special  results.  Brief  men- 
tion will  be  made  of  several  other  types  of  instruments. 

Probably  the  first  useful  application  of  harmonic  analysis 
was  to  tidal  analyzing  and  predicting  machines  by  Lord 
Kelvin,  in  1876.    The  rise  and  fall  of  the  tides,  having  been 


Fig.  101.    Kelvin's  title  predictor. 


observed  at  a  given  port  for  a  year  or  more,  is  represented 
by  a  curve  which  is  then  analyzed.  The  tidal  components 
being  known,  it  is  possible  to  synthesize  these  for  future 
dates,  that  is,  to  predict  the  tides.^^  Kelvin's  analyzer  is 
shown  in  Fig.  100,  and  the  predictor  in  Fig.  101. 

A  tide-predicting  machine  of  remarkable  completeness 
and  perfection  has  recently  been  constructed  by  the  United 
K  129 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


States  Coast  and  Geodetic  Survey  at  Washington;  Fig. 
102  is  a  general  view  of  this  instrument.'^ 

Professor  A.  A. 
Michelson  has  de- 
vised a  very  ingen- 
ious harmonic  syn- 
thesizer and  ana- 
lyzer, for  eighty 
components,  which 
he  has  applied  most 
effectively  to  the 
study  of  light 
waves.^-  A  Michel- 
son  synthesizer,  of 
recent  construction, 
for  twenty  compo- 
nents, is  shown  in 
Fig.  103;  the  ma- 
chine also  serves  as 
an  analyzer.  The 
given  wave  form  is 
cut  on  the  edge  of 
a  sheet  of  card  or 
metal,  which  is  then 
applied  to  set  the 
machine ;  a  curve  is 
drawn  by  means  of 
which  the  ampli- 
tudes of  the  required 
components  may  be 
determined  in  a  manner  described  in  the  references. 

Harmonic  analyzers  are  employed  in  electrical  engineer- 

131 


Fig.  103.    Michelson's  harmonic  analyzer  and  syn- 
thesizer for  twenty  components. 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


ing  for  the  study  of  alternating-current  waves  and  other 
periodic  curves.  The  curves  being  investigated  often  have 
few  components,  or  the  interest  is  centered  in  a  few  com- 
ponents ;  in  such  cases  simpler  forms  of  analyzers  may  be 

used  in  which  the  in- 
tegrating is  performed 
by  a  planimeter  of  the 
ordinary  type.  Figures 
104,  105,  and  106  show 
instruments  of  this  kind 
designed  by  Rowe, 
Mader,  and  Chubb, 
respectively.^^  These 
machines  may  be  used 
with  a  wave  of  any  size, 
such  as  the  original  oscillogram ;  the  curve  is  traced  with 
the  stylus,  giving  one  component ;  by  changing  one  or  more 
gear  wheels  and  again  tracing,  another  component  is 
found,  and  so  on. 


Fig.  104.    Rowe  .s  harmonic  analyzer. 


Fig.  105.    Mader's  harmonic  analyzer. 

Several  other  types  of  harmonic  analyzers  are  described 
in  Horsburgh's  ''Instruments  of  Calculation"  and  in  Morin's 
''  Les  Appariels  d'Integration."  These  books  also  describe 
many  subsidiary  instruments  and  processes  which  are  helpful 
in  numerical  work. 

132 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


Fig.  106.    Chubh's  harmonic  analyzer. 


Arithmetical  and  Graphical  ^Methods  of  Harmonic 

Analysis 

Mention  has  already  been  made  of  the  apphcation  of 
harmonic  analysis  to  the  study  of  acoustics,  the  tides,  elec- 
tricity, optics,  and  mathematics.  The  method  is  also  use- 
ful in  the  investigation  of  other  more  or  less  periodic  phe- 
nomena. In  meteorology^  it  is  applied  to  the  study  of 
hourly  or  daily  temperature  changes,  barometric  changes, 
etc.^^  In  astronomy  the  periodicity  of  sun  spots,  magnetic 
storms,  variable  stars,  etc.,  may  be  treated  by  harmonic 
analysis. In  mechanical  engineering,  valve  motions  and 
other  mechanical  movements  may  be  investigated.^^  The 
method  is  also  used  in  geophysics,  in  naval  architecture,  and 
in  the  study  of  statistics. 

When  the  number  of  curves  to  be  analyzed  is  small  and 
especially  when  the  number  of  components  is  hmited,  it 
may  not  seem  necessary  to  provide  a  machine  for  perform- 

133 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


ing  the  analyses.  While  the  general  solution  of  the  prob- 
lem was  given  by  Fourier  in  his  original  work  La  Theorie 
Analytique  de  la  C'haleur  "  (Paris,  1822),^^  yet  the  labor  in- 
volved in  the  numerical  reduction  is  very  great.  Many 
arithmetical  and  graphical  schemes  for  facilitating  the  w^ork 
have  been  developed  by  Wedmore,  Clifford,  Perry,  Kintner, 
Steinmetz,  Rosa,  Runge,  Grover,  S.  P.  Thompson,  and 
others.  In  general  a  set  of  coordinates  of  the  curve  is 
measured,  and  these  measures  are  reduced  in  accordance 
wdth  the  scheme  selected  to  give  the  amplitudes  and  phases 
of  the  components. 

Steinmetz  gives  general  formulae  systematically  arranged 
for  the  calculation  of  any  number  of  components,  of  odd 
or  even  order. Several  numerical  examples  are  given, 
selected  from  electrical  engineering,  while  another  is  the 
determination  of  the  first  seven  components  of  a  diagram  of 
mean  daily  temperatures. 

Runge's  method  depends  upon  a  scheme  of  grouping  the 
terms  so  as  to  facilitate  the  numerical  work.^^  A  number 
of  ordinates  of  the  curve,  n,  are  measured.  For  odd  com- 
ponents only,  the  ordinates  are  evenly  distributed  over  a 
half  wave  and  give  i  n  components ;  for  odd  and  even 
components  the  ordinates  are  evenly  distributed  over  the 
whole  wave,  and  give  -  1  components.  Runge  gives 
schemes  for  12,  18,  and  36  ordinates.  Bedell  and  Pierce 
give  a  scheme  and  an  example  for  determining  the  odd  com- 
ponents from  18  ordinates.^^  Carse  and  Urquhart  give  the 
scheme  with  numerical  examples  for  odd  and  even  com- 
ponents from  24  ordinates.^^  F.  W.  Grover  gives  six 
schedules  according  to  Runge's  method  wdth  examples  for 
the  calculation  of  the  odd  components  from  6,  12,  and  18 
ordinates ;  and  also  schedules  for  both  odd  and  even  com- 

134 


ANALYSIS  AND  SYNTHESIS  OF  HARMONK^  CI  RVES 


ponents  from  6,  12,  and  18  ordinates.^-  There  is  also  given 
a  special  multiplication  table  for  all  the  sine  and  cosine 
products  required.  When  one  of  these  schedules  is  appli- 
cable, the  method  as  described  by  Grover  is  probably  the 
most  expeditious  available  for  numerical  analysis.  H.  0. 
Taylor  has  developed  a  convenient  method  for  constructing 
complete  schedules  adapted  to  general  or  special  condi- 
tions.^- 

S.  P.  Thompson  has  provided  several  schedules  also  based 
on  Runge's  method,  which  are  very  expeditious.  ]\Iore 
recently  he  has  developed  an  approximate  method  of  har- 
monic analysis  in  which  all  multiplication  by  sines  and 
cosines  is  dispensed  with,  and  only  a  few  additions  and  sub- 
tractions of  the  numerical  values  of  the  ordinates  is  re- 
quired.The  method  is  applicable  only  to  periodic  curves 
in  which  the  components  higher  than  those  being  calculated 
are  absent ;  if  higher  components  are  present,  their  values 
may  be  added  to  those  of  the  lower  components  in  certain 
cases.  Thompson  gives  schedules  for  the  first  three  com- 
ponents, suitable  for  the  analysis  of  valve  motions,  a 
schedule  for  the  first  seven  components,  and  one  for  the 
odd  components  to  the  ninth,  and  a  special  schedule  suit- 
able for  tidal  analysis. 

A  large  number  of  graphical  methods  for  harmonic  analysis 
have  been  de\dsed.  These  are  suitable  for  curves  having 
only  a  few  components,  but  it  is  doubtful  whether  they  are 
any  more  expeditious  than  the  equivalent  arithmetical 
methods,  and  usually  they  are  not  so  precise.  A  convenient 
graphical  method  is  that  devised  by  Perry. Numerous 
other  methods  are  described  in  ''^Modern  Instruments  of 
Calculation"  and  in  the  volumes  of  the  Electrician.^'' 

For  comparison  the  curve  which  was  analyzed  by  machine 

135 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


in  13  minutes,  as  described  on  page  122,  was  analyzed  by 
Steinmetz's  method,  requiring  about  10  hours  to  obtain 
ten  components,  and  by  Grover's  method  the  time  was 
about  3  hours  for  eight  components  odd  and  even  (the 
largest  number  for  which  a  scheme  is  arranged).  (Grover 
mentions  that  by  his  method  eight  odd  components  can  be 
determined  in  less  than  an  hour  by  one  familiar  with  the 
process.)  The  curve  shown  in  Fig.  76,  page  100,  which 
was  known  to  have  but  three  components,  w^as  analyzed  by 
Thompson's  short  method,  the  three  amplitudes  and  phases 
being  evaluated  in  fifteen  minutes.  The  analysis  of  the 
same  curve  (for  five  components)  by  machine  required  less 
than  seven  minutes. 

Analysis  by  Inspection 

A  familiarity  with  the  effects  produced  by  various  har- 
monics on  the  shape  of  a  wave  will  often  enable  one  to 
judge  by  inspection  what  harmonics  are  present. 


Fig.  107.    Symmetrical  wave  form  of  an  electric  alternating  current. 

If  a  wave  consists  of  alternate  half-waves  which  are 
exactly  of  the  same  shape  but  opposite  in  direction,  that 
is,  if  the  wave  is  a  symmetrical  one  with,  respect  to  its 
axis,  it  can  contain  only  odd-numbered  components ;  if 

136 


ANALYSIS  AND  SYNTHESIS  OF  HARMONIC  CURVES 


a  wave  is  not  symmetrical,  it  must  contain  some  even- 
numbered  components,  and  it  may  contain  both  odd  and 
even.  Sound  waves  belong  to  the  latter  class,  no  in- 
stance having  been  observed  of  a  symmetrical  sound 
wave,  except  that  of  a  tuning  fork  which  has  only  one 
component  and  is  a  simple  sine  curve.  Electric  alter- 
nating-current waves  are  usually  of  the  first  kind,  con- 
taining only  odd-numbered  components ;  such  a  wave  is 
shown  in  Fig.  107. 

In  some  instances  a  particular  high  partial  may  be  promi- 
nent and  so  impress  its  effect  on  the  wave  as  to  produce 


Fig.  108.    Photograph  of  the  vowel  a  in  father,  intoned  upon  the  pitch  n  =  159. 


distinct  wavelets  or  ripples  on  the  main  wave  form  even 
though  this  is  itself  irregular ;  the  order  of  such  a  partial  is 
at  once  determined  by  merely  counting  the  number  of  such 
wavelets  occurring  in  one  fundamental  wave  length.  Fig. 
108  shows  a  wave  for  the  vowel  a  in  father ;  this  curve  is 
evidently  complex,  but  there  are  six  distinct  sub-peaks  on 
one  wave,  and  the  sixth  partial  is  prominent.  Since  the 
frequency  of  the  fundamental  is  known  to  be  159,  that  of 
the  sixth  partial  is  954.  Analysis  shows  that  the  first  ten 
components  of  this  curve  have  the  following  amplitudes, 
corresponding  to  a  wave  length  of  400 : 

137 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


1=4  II  =  15  III  =  18  IV  =12  V  =  20 
VI  =  60      VII  =  21      VIII  =16      IX  =   2      X  =  3 

An  instance  where  analysis  by  inspection  is  sufficient  is 
given  in  Fig.  136,  on  page  187,  which  shows  the  wave  from 
a  tuning  fork  having  but  one  overtone.  The  partial  is 
prominent  and  produces  very  definite  sub-peaks ;  it  is 
found  that  there  are  about  twenty-five  wavelets  to  four 
large  waves,  that  is,  the  partial  has  a  frequency  about  6.25 
times  that  of  the  fundamental  and  is  inharmonic. 


Fig.  109.    Clarinet  wave  showing  beats  produced  by  the  higher  partials. 


In  some  instances  the  peaks  due  to  the  partials  are  very 
pronounced  in  portions  of  a  wave  and  almost  disappear  in 
other  parts ;  this  indicates  that  there  are  beats  between 
certain  partials.  If  there  is  one  beat  per  wave  length,  it  is 
produced  by  two  adjacent  partials;  if  there  are  two  beats, 
then  the  orders  of  the  partials  differ  by  two.  The  average 
distance  between  sub-peaks  is  found  by  measurement  and, 
when  compared  with  the  wave  length,  gives  the  average 
order  of  the  prominent  partials,  from  which  it  is  then 
usually  possible  to  specify  their  exact  orders.  While  this 
method  is  most  useful  for  waves  having  a  few  components, 
such  as  alternating-current  waves,  it  may  be  applied  to  a 


138 


AXATASIS  AND  SYNTHESIS  OF  HARMONIC^  (  URVES 


wave  as  complex  as  that  of  the  tone  of  a  clarinet,  Fig.  109. 
Since  there  is  one  beat  per  wave,  there  are  tw^o  prominent 
adjacent  overtones.  Actual  measurement  of  these  wavelets 
shows  an  average  length  of  3f  millimeters  ;  the  fundamental 
.measures  38  millimeters,  about  11^  times  the  length  of  the 
sab-wave ;  the  conclusion  is  that  the  eleventh  and  twelfth 
partials  are  producing  the  beat.  The  correctness  of  this 
conclusion  is  proved  by  the  actual  analysis  which  gives  the 
follo^dng  values  for  the  first  twelve  components  of  the  curve, 
corresponding  to  a  wave  length  of  400  : 

I  =  29       II  =  7   III  =  20   IV  =  1     V  =  2      VI  =  6 
VII  =  6     VIII  =  8    IX  =  16    X  =  9   XI  =  30   XII  =  35 


A  photograph  of  the  sound  of  the  explosion  of  a  sky- 
rocket in  a  Fourth  of  July  celebration  is  shown  in  Fig.  110. 


Fig.  110.    Photograph  of  the  sound  of  the  explo.sion  of  a  skyrocket. 


The  rocket  was  about  a  quarter  of  a  mile  from  the  recording 
apparatus,  the  sound  entering  the  laboratory  through  two 
open  windows.  While  this  curve  as  a  whole  is  not  periodic, 
yet  two  or  more  periodicities  are  clearly  showm.  The  time 
signals  are  second  apart,  and  comparison  shows  one 
frequency  of  about  130  per  second  producing  the  principal 
feature  of  the  curve  :  superposed  upon  this  is  a  much  higher 
frequency  of  about  2000  per  second. 

139 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Periodic  and  Non-periodic  Curves 

The  Fourier  analysis  is  suitable  and  complete  for  any 
curve  whatever  within  the  distance  called  one  wave  length, 
even  though  there  is  no  repetition  of  this  form ;  if  the  por- 
tion analyzed  is  successively  and  exactly  repeated,  that  is, 
if  the  curve  is  periodic,  it  represents  a  wave  motion  and  the 
analysis  represents  the  entire  wave.  A  periodic  wave  is 
shown  in  Fig.  Ill,  which  is  a  photograph  of  the  vowel  a  in 
mat. 

If  a  curve  representing  some  physical  phenomenon  is 
periodic,  then  each  separate  term  of  the  Fourier  equation 


Fig.  111.    A  periodic  curve  ;  a  photograph  of  the  vowel  a  in  mat. 


of  the  curve  may  be  presumed  to  correspond  to  something 
which  has  a  physical  existence ;  it  is  the  belief  in  this  state- 
ment, amply  supported  by  investigation,  which  leads  one 
to  analyze  sound  waves  by  this  method ;  as  explained 
under  Ohm's  Law,  page  62,  each  term  is  presumed  to  cor- 
respond to  a  simple  partial  tone  which  actually  exists. 

If  the  curve  representing  the  physical  phenomenon  is 
non-periodic,  any  portion  of  the  curve  may  be  analyzed, 
and  it  will  be  completely  represented  as  to  form  by  the 
Fourier  equation,  within  the  limits  analyzed,  but  not  beyond 
these  limits.  In  this  case,  the  separate  terms  of  the  Fourier 
series  may  not  correspond  to  anything  having  a  separate 
physical  existence ;  in  fact  the  equation  may  be  presumed 

140 


ANALYSIS  AND  SYNTHf:SIS  OF  HARMONIC  CURVES 


to  be  merely  an  artificial  mathematical  formula  for  the 
short  irregular  line  which  has  been  analyzed.  The  analysis 
of  the  profile  portrait,  described  on  page  119,  illustrates 
this  application  of  Fourier  analysis  ;  there  is  no  periodicity 
of  the  wave  form,  and  the  separate  terms  of  the  equation 
can  have  no  real  significance. 

There  is  no  general  method  for  analyzing  non-periodic 
curves,  that  is,  for  curves  containing  incommensurable  (in- 
harmonic) or  variable  components  ;  such  a  method  is  very 
much  desired  for  the  study  of  noises  and  of  sounds  from 
such  sources  as  bells,   whispered  words,   the  consonant 


Fig.  112.    A  non-periodic  curve:  a  photograph  of  the  sound  from  a  bell. 


sounds  of  speech,  and  in  fact  all  speech  sounds  except  the 
simple  vowels ;  this  need  is  probably  the  greatest  of  those 
unpro\4ded  for.-^^-^^  A  non-periodic  curve,  a  photograph 
of  the  sound  from  a  bell,  is  shown  in  Fig.  112  ;  there  is  no 
apparent  wave  length  in  this  curve,  and  an  analysis  of  any 
portion  of  it  would  probably  give  an  equation  containing 
an  infinite  number  of  terms,  though  the  real  sound  is  un- 
doubtedly compounded  from  a  finite  number  of  partials 
which  are  inharmonic  and  therefore  indeterminate.  ]\Iuch 
information  may  be  obtained  from  such  curves  by  making 
skillfully  assumed  analyses. 


141 


LECTURE  V 


INFLUENCE  OF  HORN  AND  DL\PHRAGM  ON  SOUND 
WAVES,  CORRECTING  AND  INTERPRETING  SOUND 
ANALYSES 

Errors  in  Sound  Records 

The  photographs  and  analyses  of  sound  waves  obtained 
by  the  compUcated  mechanical  and  numerical  processes 
described  in  Lectures  III  and  IV,  are,  unfortunately,  not 
yet  in  suitable  form  for  determining  the  tone  characteristics 
of  the  sounds  which  they  represent.  Before  these  analyses 
can  furnish  accurate  information  they  must  be  corrected  for 
the  effects  of  the  horn  and  the  diaphragm  of  the  recording 
instrument,  a  correction  involving  fully  as  much  labor  as 
was  expended  on  the  original  work  of  photography  and 
analysis.  For  the  sake  of  greater  emphasis,  it  may  be  di- 
rectly stated  that  the  neglect  of  the  corrections  for  horn 
and  diaphragm  often  leads  to  wholly  false  conclusions  regard- 
ing the  characteristics  of  sounds,  since  horns  and  diaphragms 
of  different  types  give  widely  differing  curves  for  precisely 
the  same  sound. 

For  research  upon  complex  sound  waves,  a  recording 
instrument  using  a  diaphragm  should  possess  the  following 
characteristics :  (a)  the  diaphragm  as  actually  mounted 
should  respond  to  all  the  frequencies  of  tone  being  investi- 
gated ;  (6)  it  should  respond  to  any  combination  of  simple 
frequencies ;  (c)  it  should  not  introduce  any  fictitious  fre- 
quencies; (d)  the  recording  attachment  should  faithfully 

142 


1 

ERRORS  IN  SOUND  RECORDS 


transmit  the  movements  of  the  diaphragm ;  and  (e)  there 
must  be  a  determinate,  though  not  necessarily  simple, 
relation  between  the  response  to  a  sound  of  any  pitch  and 
the  loudness  of  that  sound. 

It  is  well  known  that  the  response  of  a  diaphragm  to 
waves  of  various  frequencies  is  not  proportional  to  the 
amplitude  of  the  wave ;  the  diaphragm  has  its  own  natural 
periods  of  vibration,  and  its  response  to  impressed  waves 
of  frequencies  near  its  own  is  exaggerated  in  degrees  de- 
pending upon  the  damping.  The  resonating  horn  also 
greatly  modifies  waves  passing  through  it.  Therefore,  it 
follows  that  the  resultant  motion  of  the  diaphragm  is  quite 
different  from  that  of  the  original  sound  wave  in  the  open  air. 

The  theory  of  these  disturbances  for  simple  cases  is 
complete,  but  what  actually  happens  in  a  given  practical 
apparatus  is  made  indeterminate  by  conditions  which  are 
complicated  and  frequently  unknown.  There  being  no 
available  solution  of  this  problem,  it  was  necessary  to  make 
an  experimental  study  of  these  effects  as  they  occur  in  the 
phonodeik.^^ 

It  has  been  proved  that  the  phonodeik  possesses  several 
of  the  characteristics  mentioned ;  (a)  it  has  been  shown  by 
actual  trial  that  it  responds  to  all  frequencies  to  12,400 ; 
(b)  various  combinations  of  simple  tones  up  to  ten  in  num- 
ber have  been  actually  produced  with  tuning  forks,  and 
the  photographic  records  have  been  analyzed ;  (c)  in  each 
case  the  analysis  shows  the  presence  of  all  tones  used,  and 
no  others.  We  have  then  only  to  determine  the  accuracy 
with  which  the  response  represents  the  original  tanes,  the 
qualifications  {d)  and  (e)  mentioned  above ;  this  requires 
the  investigation  of  all  the  factors  of  resonance,  interfer- 
ence, and  damping. 

143 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Ideal  Response  to  Sound 

In  Lecture  II  it  has  been  explained  that  the  intensity  or 
loudness  of  a  simple  sound  is  proportional  to  the  square  of 
the  amplitude  multiplied  by  the  square  of  the  frequency, 
that  is,  to  {nAy.  A  recording  apparatus  having  ideal 
response  must  fulfill  the  following  condition :   let  all  the 


C2  D    EF    G    A    BC3  D    EF    G    A    BC4   D    EF    G    A    BC5  D    EF    C    A    BCg  D    EF    G    A  BC7 


129 


2069 


Fig.  113.    Curve  showing  the  amplitudes  for  a  sound  of  varying  pitch  but  of 

constant  loudness. 

tones  of  the  musical  scale  (simple  tones)  from  the  lowest 
to  the  highest,  and  all  exactly  of  the  same  loudness,  be  sounded 
one  after  the  other  and  be  separately  recorded ;  let  the 
amplitudes  of  the  various  responses  be  measured,  and  each 
amplitude  be  multiplied  by  the  frequency  of  the  tone  pro- 
ducing it ;  then  the  squares  of  the  products  of  amplitude  and 
frequency  must  be  constant  throughout  the  entire  series. 

A  curve  of  ideal  amplitudes  is  given  in  Fig.  113,  the  verti- 
cal scale  of  which  is  one  of  linear  measure,  centimeters  for 

144 


ERRORS  IN  SOUND  RECORDS 


instance,  and  the  horizontal  scale  is  a  logarithmic  scale  of 
frequencies.  The  di\4sions  represented  by  the  light  verti- 
cal lines  correspond  to  the  successive  tones  of  the  musical 
scale,  as  is  explained  later  in  this  lecture.  The  properties 
of  this  curve  are  as  follows  :  if  a  simple  sound  having  the 
pitch  C3  =  259  produces  a  record  which  has  an  amplitude 
represented  by  the  ordinate  a  of  the  curve,  then  a  sound 
of  exactly  the  same  loudness,  but  one  octave  lower  in  pitch, 
is  correctly  represented  by  a  record  the  amplitude  of  which 
is  the  ordinate  h;  further,  the  sound  A4,  having  870  vibra- 
tions per  second,  and  of  the  same  loudness  as  either  of 
the  others,  should  produce  an  amplitude  measured  by  the 
much  shorter  ordinate,  c:  and  similarly  for  any  note  of  the 
scale. 

Actual  Response  to  Sound 

The  determination  of  the  actual  response  of  a  recording 
apparatus  requires  a  set  of  standards  of  tone  intensity  for 
the  entire  scale  of  frequencies  under  investigation.  The 
practical  fulfillment  of  this  requirement  for  a  time  seemed 
an  impossibility.  A  manufacturer  of  organ  pipes  who 
became  interested  in  the  problem  provided  two  complete 
sets  of  pipes,  an  open  diapason  of  metal  and  a  stopped  dia- 
pason of  wood,  especially  voiced  and  regulated  to  uniform 
loudness  throughout,  according  to  his  skilled  judgment. 
The  stopped  diapason  pipes.  Fig.  114,  sixty-one  in  number, 
range  in  pitch  from  C2  =  129  to  C7  =  4138 ;  the  scale  is 
extended  by  nineteen  specially  voiced  metal  pipes  to  a  pitch 
12,400.  The  adjustment  of  these  pipes  for  uniform  loud- 
ness has  been  improved  and  verified  by  two  experimental 
methods.  While  this  scale  is  arbitrary  and  of  moderate 
precision,  it  is  the  only  available  method  by  which  progress 
has  been  possible,  and  its  use  has  led  to  most  interesting 
L  145 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


results  in  studying  the  responses  and  correction  factors  of 
the  phonodeik  under  various  conditions. 

The  sounds  from  the  several  pipes  of  the  two  sets  have 
been  photographed  and  analyzed.  The  analyses  show  that 
the  open  diapason  pipes  have  a  strong  octave  accompanying 
the  fundamental,  while  the  stopped  pipes  give  practically 
simple  tones  ;  the  latter  are  used  exclusively  in  obtaining 
the  correction  factors  as  explained  later. 

These  pipes  are  sounded  in  front  of  the  phonodeik,  one 


lilil<Nlil<!<lll<liliy<liliilH!i!iJilM 


Fig.  114.    A  set  of  organ  pipes  of  uniform  loudness. 

at  a  time,  and  the  resulting  ampUtudes  of  vibration  of  the 
diaphragm  are  recorded  photographically.  The  film  is 
stationary ;  the  first  pipe,  Co,  is  sounded  steadily ;  the 
shutter  is  released,  giving  an  exposure  of  about  second  ; 
the  spot  of  light  which  is  vibrating  back  and  forth  in  a 
straight  line  falls  on  the  film,  making  several  excursions 
mthin  the  time  of  exposure,  and  records  the  amplitude  of 
the  vibration ;  the  record  for  the  first  pipe  is  C  in  Fig. 
115.  The  film  is  moved  lengthwise  about  a  quarter  of  an 
inch,  and  while  the  second  pipe,  C 2 if,  is  sounding,  the  result- 
ing amplitude  is  photographed.    The  process  is  continued 

146 


ERRORS  IN  SOUND  RECORDS 


until  the  amplitudes  produced  by  the  sixty-one  pipes  are 
recorded.  The  vertical  scale  of  such  a  chart  represents 
linear  amplitude,  while  the  horizontal  scale  is  a  logarithmic 
scale  of  frequencies  which  is  described  on  page  168. 

A  curve  may  be  drawn  through  the  upper  ends  of  these 
amplitude  records,  showing  the  "  responsivity "  of  the 
apparatus  under  the  conditions  of  the  experiment.  Fig. 
115  shows  the  responses  used  in  correcting  the  analysis  of 
the  organ-pipe  curve  shown  on  page  122,  while  the  inter- 
pretation of  the  responses  is  given  on  page  163. 


c  .        ,^  :'' 
J. .... 

1  ' 

1 

 |-|-[|-|-I--H-I4--H-|4|^ 

pi.' 

1 

|....|.l.|.|.H.i. 

Fig.  115.    Photographic  record  of  the  ampHtudes  of  vibration  for  the  organ  pipes 

of  uniform  loudness. 


The  irregular  curve  of  Fig.  116  is  the  response  obtained 
with  one  of  the  earliest  forms  of  phonodeik ;  it  shows  an 
almost  startling  departure  from  the  ideal  response  repre- 
sented by  the  smooth  curve.  What  produces  the  range 
of  mountains,  with  sharp  peaks  and  valleys  ?  Why  is  there 
no  response  for  the  frequency  1460 ;  why  is  it  excessive  for 
frequencies  from  2000  to  3000?  There  were  five  suspected 
causes :  (1)  unequal  loudness  of  the  pipes ;  (2)  the  dia- 
phragm effects ;  (3)  the  mounting  and  housing  of  the  dia- 
phragm ;  (4)  the  vibrator  attached  to  the  diaphragm ;  (5) 
the  horn. 

147 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


The  investigation  of  these  pecuUarities  was  most  tantahz- 
ing ;  the  peaks  acted  hke  imps,  jumping  about  from  place 
to  place  with  every  attempt  to  catch  them,  and  chasing 
and  pushing  one  another  in  a  very  exasperating  manner. 
Perhaps  two  months'  continuous  search  was  required  to 
find  the  causes  of  ^^1460"  and  '^2190"  alone.  The  investi- 
gations led  to  many  improvements  in  the  phonodeik  and  to 


129     .-ssTJ:j:n^     259  517  1035  2069  4138 


Fig.  116.    A  response  curve  obtained  with  an  early  form  of  phonodeik. 

a  practical  method  of  correction  for  the  departures  from 
ideal  response. 

Response  of  the  Diaphragm 

Experiments  have  been  made  with  diaphragms  of  several 
sizes  and  thicknesses,  and  of  various  materials,  such  as 
iron,  copper,  glass,  mica,  paper,  and  albumen.  The  experi- 
ments described  in  this  section  concern  circular  glass  dia- 
phragms ha\dng  a  thickness  of  0.08  millimeter,  and  held 
around  the  circumference,  either  firmly  clamped  between 
hard  cardboard  gaskets  and  steel  rings,  or  loosely  clamped 

148 


INFLUENCE  OF  DIAPHRAGM  ON  SOUND  RECORDS 


between  soft  rubber  gaskets ;  the  diaphragm  is  entirely 
free,  there  being  no  horn  or  housing  of  any  kind.  The  silk 
fiber  of  the  phonodeik  vibrator  is  attached  to  the  diaphragm 

! 


C2  D    EF    G    A  ^  B^^  BC4   D    EF    G    A    BC5  D    EF    G    A    BCe  D    EF    OA  BC7 

'  |JN^I^iyi?N|sNg|»IM8|iW»|i|»N|sNi;ij5i;i;|^ 

129  259  517  1035  2069  4138 

Fig.  117.    Resonance  peaks  for  diaphragms  of  different  diameters. 

to  record  its  movements.  The  pipes  already  described 
were  sounded  in  succession  in  front  of  the  diaphragm,  and 
observations  were  made  of  the  response  under  various 

149 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


conditions.  The  responses  of  three  glass  diaphragms  of 
22,  31,  and  50  miUimeters  diameter,  respectively,  are  shown 
in  a,  h,  and  c,  Fig.  117. 

When  the  pitch  of  the  pipe  being  sounded  is  near  the 
natural  frequency  of  the  diaphragm,  the  latter  moves 
easily  and  responds  vigorously ;  the  diaphragm  is  in  sym- 


a 


h 


c 


1 

\ 

1 

7 

7 

\ 

] 

\ 

9 

e 

176 

J 

^ 

/ 

/ 

r 

\ 

D 

\ 

4 

0 

17& 

C,  D   EF    G    A    BC3  D    EF    G   A    BC4   D    EF    G    A    BCj  D    EF    G    A    BCg  D    EF    G   A  BC7 

i;Miki^i^biii.i^i5iii;asiiisfefeiiiti;i;iii^^^ 

129  259  517  1035  2069  4138 

Fig.  118.    Effects  of  clamping  and  distance  cm  the  response  of  a  diaphragm. 

pathy  or  in  resonance  with  the  tone ;  the  response  curve 
shows  a  peak  for  such  a  resonance  condition.  The  natural 
period  of  the  largest  diaphragm  had  a  frequency  of  366, 
corresponding  to  which  there  was  a  large  response,  as  shown 
in  the  lower  curve  of  the  figure.  Two  other  peaks  repre- 
sent the  natural  overtones  of  the  diaphragm ;  these  over- 
tones have  frequencies  3.28  and  6.72  times  that  of  the  funda- 
mental, and  are  inharmonic.    The  other  curves  show  that 

150 


INFLUENCE  OF  DIAPHRAGM  ON  SOUND  RECORDS 


the  natural  period  of  the  diaphragm  rises  in  pitch  as  the 
diameter  decreases,  and  that  the  actual  response  becomes 
less. 

The  lower  curve,  c,  Fig.  118,  is  the  response  of  a  glass 
diaphragm  31  millimeters  in  diameter,  held  lightly  in  the 
clamping  rings.  When  the  clamping  is  tightened,  the 
response  is  as  shown  in  the  middle  curve  ;  the  natural  period 
is  increased  from  640  to  916,  while  the  amplitude  is  reduced 
from  228  to  160.  The  upper  curve  shows  the  response  for 
the  same  diaphragm  as  for  the  middle  curve,  but  \vith  the 
pipes  at  a  greater  distance  ;  the  curve  is  of 'the  same  general 
shape,  while  the  response  is  of  diminished  amplitude. 

Chladni's  Sand  Figures 

Chladni's  method  of  sand  figures  has  been  employed  in 
stud^dng  the  conditions  of  \dbration  of  the  diaphragm.^" 
A  plate  or  diaphragm,  clamped  at  the  edge  or  at  an  interior 
point,  may  be  made  to  vibrate  in  many  different  modes. 
When  sand  is  strewn  on  the  plate  it  is  observed  that  por- 
tions are  moving  up  and  down,  throwing  the  sand  into  the 
air.  There  are  certain  hues  toward  which  the  sand  gathers, 
indicating  that  these  parts  are  relatively  at  rest ;  the  lines 
on  w^hich  the  sand  accumulates  are  called  nodal  lines,  and 
form  patterns  or  figures  which  are  always  the  same  for  the 
same  note,  but  differ  for  each  change  of  pitch  or  quality. 
It  is  thus  shown  that  a  diaphragm  \dbrates  in  various 
subdi\dsions. 

A  diaphragm  of  glass  held  in  circular  rings  was  placed 
horizontally,  the  vibrator  being  attached  to  the  under  side  ; 
sand  was  then  sprinkled  over  the  diaphragm  which  was  made 
to  respond  in  succession  to  each  one  of  the  eighty  pipes  of 
frequencies  from  129  to  12,400.    The  characteristic  nodal 

151 


THE  SCIENCE  OF  MUSICAL  SOUNDS 

lines  produced  in  each  instance  were  either  sketched  or 
photogi'aphed. 

When  the  test  sound  has  a  pitch  equal  to  the  natural 
frequency  of  the  diaphragm,  366,  the  diaphragm  ^dbrates 


Fig.  119.    Different  modes  of  vibration  of  a  diaphragm,  shown  by  sand  figures. 


as  a  whole  \dgorously ;  there  are  no  nodal  lines,  except  the 
circumference,  and  even  this  is  probably  in  motion  when  the 
clamping  is  light.  There  are  no  nodal  hues  for  tones  within 
the  octave  lower  and  the  octave  higher  than  the  natural 
frequency. 

152 


INFLUENCE  OF  DIAPHRAGM  ON  SOUND  RECORDS 


As  the  pitch  of  the  test  sound  rises,  the  area  of  the  plate 
which  can  respond  seems  to  be  less  than  the  whole,  and  this 
part  moves,  with  the  formation  of  a  nodal  boundary  hne 
of  more  or  less  irregular  shape  ;  Fig.  119,  a,  is  the  photograph 
of  the  pattern  for  the  frequency  977.  Since  parts  of  the 
plate  are  now  at  rest,  no  part  can  ^dbrate  through  a  large 
amphtude,  and  the  response  is  greatly  diminished,  as  shown 
in  the  response  curve.  Fig.  117.  A  second  maximum  re- 
sponse is  obtained  from  a  sound  corresponding  to  the 
first  overtone  of  the  diaphragm  ha\dng  a  frequency  of 
1200,  about  3.28  times  that  of  the  fundamental ;  this  is 
represented  by  the  second,  smaller  peak  of  the  curve. 
The  nodal  figure  on  the  diaphragm  is  a  circle  of  medium 
size. 

As  the  pitch  of  the  test  sound  rises,  the  figures  again 
become  irregular  and  of  smaller  area,  and  two  concentric 
nodal  circles  appear,  b,  Fig.  119,  corresponding  to  the  second 
overtone,  of  a  frequency  of  2460,  6.72  times  that  of  the 
fundamental.  As  the  pitch  rises  stiU  higher,  the  areas 
become  smaller,  with  the  formation  of  three,  four,  and  five 
concentric  circles,  and  other  designs.  The  photographs  c 
and  d  show  the  nodal  lines  for  frequencies  of  2600  and  10,400. 

Free  Periods  of  the  Diaphragm 

Besides  the  two  methods  already  described,  one  by  direct 
measure  of  the  response,  the  second  by  means  of  the  Chladni 
sand  figures,  a  third  method  of  determining  the  diaphragm 
characteristics  has  been  used,  that  of  photographing  the 
free-period  effects.  The  diaphragm  is  given  a  single  dis- 
placement, and  upon  release  is  allowed  to  vibrate  freely. 
This  displacement  may  be  produced  by  the  noise  of  a  hand- 
clap, or  by  attaching  a  fine  thread  to  the  diaphragm  which 

153 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


is  gently  pulled  aside  and  the  thread  then  burned.  The 
frequency  is  given  by  comparison  with  the  time  signals. 

The  curve  a,  Fig.  120,  was  obtained  with  a  free,  uncov- 
ered diaphragm ;  two  distinct  frequencies  are  shown  at  the 
beginning  of  the  motion,  1000  and  3100 ;  the  latter  persists 
foi"  2^0  second,  while  the  former  lasts  about  q^j  second. 
Curve  h  was  obtained  when  the  diaphragm  was  inclosed  in 
a  housing  forming  a  front  and  back  cavity,  but  with  no 
horn ;  the  air  cushions  damp  the  \dbrations,  there  being 


Fig.  120.    Free-period  \abrations  of  a  diaphragm. 


nine  vibrations  now,  while  before  there  were  twenty-two ; 
the  frequency  of  the  diaphragm  has  been  reduced  to  400, 
and  there  is  a  higher  frequency  of  2190  due  probably  to  the 
natural  period  of  the  air  in  the  chambers.  When  a  horn  is 
added,  the  curve  c  is  obtained ;  the  frequency  of  vibration 
of  the  air  in  the  horn  is  264,  and  it  continues  to  vibrate  for 
about  a  tenth  of  a  second ;  the  frequency  of  the  diaphragm 
is  now  530,  and  that  of  the  back  cavity  is  2190 ;  these  vi- 
brations die  out  in  about     o  second,  as  before. 

154 


INFLUENCE  OF  DIAPHRAGM  ON  SOUND  RECORDS 


Influence  of  the  Mounting  of  the  Diaphragm 


In  the  early  experiments  it  was  thought  desirable,  in 
order  to  protect  the  diaphragm  from  indirect  sounds,  to 
inclose  it  in  a  housing ;  various  shapes  and  sizes  of  front 
and  back  coverings,  shown  in  Fig.  121,  were  tried.  The 
diaphragm  is  in  effect  between  two  cavities,  and  it  was 
found  that  each  produces  its  own  complete  and  independent 
resonance  effects,  and  that  these  influence  each  other  through 
the  diaphragm.  When  the  frequencies  of  these  cavities 
are  in  certain 
ratios,  the  re- 
sponse of  the 
diaphragm  is 
annulled  by  in- 
terference ef- 
fects ;  at  other 
times  these 
cavities  pro- 
duce exagger- 
ated responses. 

Experiment  indicated  that  the  back  should  be  uncovered, 
since  the  effect  of  a  sound  produced  in  front  of  the  horn  is 
ordinarily  of  no  influence  on  the  back  of  the  diaphragm. 
The  front  must,  of  course,  be  covered  and  the  connections 
between  the  cover  and  the  horn,  and  the  cover  and  the  dia- 
phragm, must  be  air-tight.  The  best  results  were  obtained 
by  using  a  shallow  cup-shaped  front  cover  with  an  opening 
for  the  horn,  which  may  have  a  diameter  about  one  fourth 
of  that  of  the  diaphragm.  If  the  front  cover  is  close  to  the 
diaphragm,  the  damping  effect  of  the  air  cushion  may  be 
too  great. 

155 


Fig.  121. 


Shapes  of  front  and  back  coverings  for  a 
diaphragm. 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Influence  of  the  Vibrator 

Elaborate  studies  have  been  made  of  the  influence  of  the 
vibrator  on  the  response  of  the  apparatus ;  among  the 
factors  investigated  are  the  mass  and  shape  of  the  vibra- 
tor, size  and  shape  of  the  mirror,  material  and  length  of 
the  fiber,  material  and  length  of  the  tension  piece  and  its 
hysteresis  and  damping  effects,  amount  of  tension,  moduli 
of  elasticity,  and  temperature  ;  computations  also  have  been 
made  of  the  inertias,  accelerations,  forces,  and  natural 
periods,  of  the  various  parts,  and  their  resonances  and  inter- 
ferences for  frequencies  up  to  10,000  ;  the  differential  equa- 
tions of  motion  of  the  actual  system  have  been  formed  and 
solved. 

It  is  quite  out  of  place  to  explain  the  details  of  this  work 
here ;  the  final  practical  result  is  the  demonstration  that 
for  frequencies  less  than  5000  the  vibrator  produces  no 
appreciable  effect  on  the  record. 

This  conclusion  is  verified  by  the  results  of  a  further 
study  with  the  Chladni  sand  figures ;  besides  the  set  of 
figures  described  previously,  with  the  vibrator  attached  to 
the  diaphragm,  a  second  complete  set  of  figures  was  ob- 
tained without  the  vibrator ;  the  two  sets  are  practically 
identical  except  for  a  shifting  of  the  nodal  lines  for  high 
frequencies. 

Influence  of  the  Horn 

A  horn  as  used  with  instruments  for  recording  and  repro- 
ducing sound  is  usually  a  conical  or  pyramidal  tube,  the 
smaller  end  of  which  is  attached  to  the  soundbox  contain- 
ing the  diaphragm,  while  the  larger  end  opens  to  the  free 
air.  The  effect  of  the  horn  is  to  reinforce  the  vibrations 
which  enter  it  due  to  the  resonance  properties  of  the  body 

156 


INFLUENCE  OF  THE  HORN  ON  SOUND  RECORDS 


of  air  inclosed  by  the  horn.  The  quantity  and  quality  of 
resonance  depends  mainly  upon  the  volume  of  the  inclosed 
air  and  somewhat  upon  its  shape.  If  the  walls  of  the  horn 
are  smooth  and  rigid,  they  produce  no  appreciable  effect 
upon  the  tone.  But  if  the  w^alls  are  rough  or  flexible,  they 
may  absorb  or  rapidly  dissipate  the  energy  of  vibrations  of 


Fig.  122.    Experimental  horns  of  various  materials,  sizes,  and  shapes. 


the  air  of  certain  frequencies  and  thus  by  subtraction  have 
an  influence  upon  tone  quality.  The  horn  of  itself  cannot 
originate  any  component  tone,  and  hence  cannot  add  any- 
thing to  the  composition  of  the  sound.  The  horn  is  an  air 
resonator  and  not  a  soundboard ;  any  vibrations  which  the 
walls  of  the  horn  may  have  are  relatively  feeble  and  are  re- 
ceived from  the  air  which  is  already  in  vibration,  while  in 
the  case  of  a  soundboard,  the  air  receives  its  vibration  from 

157 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


the  soundboard  as  a  source.  Because  the  horn  operates 
through  the  inclosed  air,  it  is  a  very  sensitive  resonator,  and 
hence  its  usefulness  when  its  action  is  understood  and 
properly  applied. 

A  horn  used  in  connection  with,  the  diaphragm  very 
greatly  increases  the  response,  but  it  also  adds  its  own  nat- 
ural-period effects,  which  are  quite  complex.  A  variety 
of  horns,  shown  in  Fig.  122,  were  used  in  the  experiments ; 
these  are  of  various  materials,  sheet  zinc,  copper,  thick  and 
thin  wood,  and  artificial  stone  ;  one  horn  was  made  \vith 
double  walls  of  thin  metal,  and  the  space  between  was 
filled  with  water.  Probabl}^  the  most  rigid  material,  such 
as  stone  or  thick  metal,  gives  the  best  results.  For  con- 
venience, however,  sheet  zinc  is  used  with  the  phonodeik, 
and  so  long  as  the  horn  is  supported  under  constant 
conditions,  which  are  involved  in  the  ''correction  curve'' 
described  later,  this  material  is  satisfactory. 

A  horn  such  as  is  used  in  these  experiments  has  its  own 
natural  tones,  which  can  be  brought  out  by  blowing  with  a 
mouthpiece  as  in  a  bugle  ;  these  tones  are  a  fundamental 
with  its  complete  series  of  overtones.  The  fundamental 
pitch  can  be  heard  by  tapping  the  small  opening  \vith.  the 
palm  of  the  hand. 

When  a  horn  is  added  to  the  diaphragm,  the  response  is 
greatly  altered ;  Fig.  1 23  shows  the  response  curves  for 
three  horns  of  different  lengths.  In  each  curve  the  peaks 
corresponding  to  the  fundamental  of  the  horn,  the  octave, 
and  the  other  overtones  up  to  the  seventh,  are  distinct. 
These  peaks  are  indicated  in  the  figure  by  Hi,  H2,  etc.,  while 
the  diaphragm  peak  is  marked  D. 

In  the  upper  curve,  the  peak  due  to  the  diaphragm  comes 
between  the  peaks  for  the  fundamental  and  the  octave  of 

158 


INFLUENCE  OF  THE  HORN  ON  SOUND  RECORDS 


the  horn ;  the  latter  peaks  are  pushed  apart,  one  being 
lowered  in  pitch  and  the  other  raised,  so  that  the  interval 
between  them  is  two  semitones  more  than  an  octave  ;  the 
peak  for  the  third  partial  is  in  its  proper  position.  The 
horn  reacts  upon  the  diaphragm,  causing  it  to  have  a  period 
different  from  that  which  it  had  before  the  horn  was  appUed. 


Fig.  123.    Resonance  peaks  for  horns  of  various  length.s. 


The  middle  curve  shows  the  response  with  a  longer  horn  ; 
the  diaphragm  peak  now  comes  between  the  peaks  of  the 
second  and  third  partials,  and  both  these  and  that  of  the 
fourth  are  displaced.  The  lower  curve  represents  the  re- 
sponse for  a  still  longer  horn. 

A  long  horn  seems  to  respond  nearly  as  well  to  high  tones 
as  does  a  short  one,  while  the  response  to  low  tones  is  much 
greater ;  the  response  below  the  fundamental  of  the  horn 
is  very  feeble.    The  horn  selected  should  be  of  such  a 

159 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


length  that  its  fundamental  is  lower  than  the  lowest  tone 
under  investigation.  For  the  study  of  vowel  sounds,  the 
horn  employed  has  a  length  of  about  48  inches,  giving  a 
fundamental  frequency  of  about  125. 

It  is  important  that  there  are  no  holes,  open  joints,  or 
leaks  of  any  kind  in  the  walls  of  the  horn,  because  tones 
with  a  node  in  the  position  of  a  hole  will  be  absent.    A  hole 


a 


/\ 


259  517  1035  2069 

Fig.  124.    Resonance  peaks  for  horns  of  various  flares. 


4138 


one  millimeter  in  diameter  is  sufficient  to  alter  the  response. 

The  flare  of  the  horn  has  a  great  influence  upon  the  re- 
sponse ;  Fig.  124  shows  the  responses  obtained  with  three 
horns  of  the  same  length,  but  of  different  flares.  The  upper 
curve  is  the  response  for  a  narrow  conical  horn,  the  large 
end  of  which  has  a  diameter  equal  to  one  fifth  of  the 
length ;  the  middle  curve  is  for  a  wide  cone,  the  diameter 
of  the  open  end  being  one  half  of  the  length  ;  the  lower  curve 

160 


INFLUENCE  OF  THE  HORN  ON  SOUND  RECORDS 

is  for  a  horn  of  flaring,  bell  shape.  Widening  the  mouth 
increases  the  effect  in  a  general  way ;  the  bell  flare  makes 
the  natural  periods  indefinite,  and  heaps  up  the  response 
near  the  fundamental,  diminishing  that  for  the  higher 
tones. 

The  shape  selected  for  use  wdth  the  phonodeik  is  a  cone 


J-l-rl-l-l°l- rrrri  J~rrrrrri''l''hl°l  J'^l-°hrl*l-'rl'°rl°I^M-l-l-l-l-l-l-l-l-l-l-l 
129  259  517  1035  20C9 


39  4138 


Fig.  125.    Resonance  effect  of  the  horn. 


of  medium  flare ;  this  gives  a  good  distribution  of  response, 
and  the  resonances  are  definite,  but  not  too  sharp  to  allow 
of  correction. 

The  curve  a,  Fig.  125,  is  an  actual  response  curve  con- 
taining both  horn  and  diaphragm  effects,  h  is  the  diaphragm 
response,  while  c  and  d  are  the  ideal  curves  pre\'iously 
explained.    The  effect  of  the  natural  period  of  the  diaphragm 

M  161 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


is  represented  by  the  sharp  peak  D.  If  the  diaphragm 
responded  ideally,  the  curve  6  would  coincide  with  c  through- 
out its  length,  and  a  would  then  indicate  the  effects  due  to 
the  horn  alone.  The  ordinates  of  the  curve  a  have  been 
multiplied  by  such  factors  as  will  reduce  the  diaphragm 
curve  to  the  ideal,  and  the  results  are  plotted  in  e.  The 
difference  between  the  curve  e  and  the  ideal  d  is  the  effect 
due  to  the  horn,  corrected  for  the  peculiarities  of  the  dia- 
phragm, in  the  manner  explained  later  in  this  lecture. 

The  line  /  indicates  the  location  of  the  natural  harmonic 
overtones  of  the  horn ;  the  curve  e  shows  by  its  peaks  that 
the  horn  strongly  reinforces  the  tones  near  its  own  funda- 
mental, and,  in  a  diminishing  degree,  those  near  all  of  its 
harmonics.  The  resonance  of  the  horn  increases  the  effects 
of  all  tones  corresponding  to  the  complete  series  of  har- 
monics which  the  horn  itself  would  give  if  it  were  blown  as 
a  bugle  or  hunting  horn. 

The  diaphragm  peak  D  comes  between  the  peaks  for  the 
third  and  fourth  partials  of  the  horn,  and  it  in  effect  divides 
the  partials  into  two  groups  which  are  pushed  apart  in 
pitch ;  the  amount  of  this  displacement  is  shown  by  the 
gap  in  the  line  /  near  the  fourth  point. 

Correcting  Analyses  of  Sound  Waves 

The  investigation  of  the  effects  of  the  horn  and  diaphragm 
of  a  sound-recording  apparatus,  a  few  details  of  which 
have  been  described,  involved  an  unexpected  amount  of 
labor  ;  it  is  estimated  that  the  time  required  was  equivalent 
to  that  of  one  investigator  working  eight  hours  for  every 
working  day  in  three  and  a  half  years.  It  has  been  shown 
that  the  horn  and  diaphragm  introduce  many  distortions 
into  the  curves  obtained  with  their  aid,  and  that  the  dis- 

162 


CORRECTING  ANALYSES  OF  SOUND  WAVES 


tortions  vary  greatly  with  the  conditions  of  the  instrument. 
Unless  these  errors  are  eliminated  and  the  true  curves 
found,  the  records  of  the  instrument  will  be  without  value 
because  the  incorrect  and  false  curves  can  lead  to  no  rational 
conclusions  whatever.  One  may  wonder,  if  the  horn  pro- 
duces such  disturbances,  why  it  is  not  dispensed  with  in 
scientific  research  ;  the  horn  has  been  retained  because  the 


i 

.-^ 

1 

2 

3 

-4 

4 

1 

s 

6 

J 
7 

8 

1 

9 

1 

1  1 

1 

1 

5 

1 

I 

20 

1 

A 

D 

,  1 

1  : 

3 

1 

EF 

G 

A   BC3  D  E 

F  G 

A 

BC4 

D  EF 

G 

A 

BCs  D  EF 

G 

?^ 

D 

EF  G 

^  1  1  .  1 

D 

E 

mm 

m 

i|i 

.1^1 

w 

l|i 

!|? 

'1. 

i|8 

i| 

129  258  517  1035  2069  4108 


Fig.  126.    Curves  used  in  correcting  analyses  of  sound  waves. 

sensitiveness  of  the  recording  apparatus  is  increased  several 
thousand-fold  by  its  use. 

Whenever  records  are  made  for  analytical  purposes,  the 
condition  of  the  phonodeik  is  usually  determined,  as  pre- 
viously explained,  by  photographing  its  response  to  each 
of  a  set  of  sixty-one  organ  pipes  of  standard  intensity, 
covering  a  range  of  frequencies  from  129  to  4138.  The 
actual  response  of  the  phonodeik  in  the  form  for  research 
is  shown  by  the  irregular  curve  a,  Fig.  126,  while  h  is  the 

163 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


desired  or  ideal  response.  The  shape  of  the  curve  will 
vary  with  every  change  in  the  size  or  condition  of  the  horn 
or  diaphragm,  with  temperature  changes,  with  the  tight- 
ness of  clamping  of  the  diaphragm,  with  the  nature  of  the 
room  in  which  the  apparatus  is  used,  and  with  other  condi- 
tions. The  sharp  resonance  peak  D  is  due  to  the  free 
period  of  the  diaphragm. 

The  ordinates  of  this  curve  give  the  amplitudes  of  the 
phonodeik  records  for  sounds  of  various  pitches,  all  being 
of  the  same  intensity.  The  curve  shows  that  the  tone  of 
any  particular  pitch,  whether  a  single  simple  tone  or  a 
single  component  of  a  complex  sound,  in  general  produces 
a  response  either  too  large  or  too  small  as  compared  with 
the  response  due  to  other  pitches.  When  any  sound  has 
been  photographed  with  the  phonodeik  in  the  conditions 
here  represented,  and  the  amplitudes  of  all  the  components 
of  the  complex  tone  have  been  determined  by  analysis,  it 
is  necessary  to  correct  each  individual  amplitude  by  multi- 
plying it  by  a  factor  corresponding  to  its  particular  pitch. 
The  factor  for  a  tone  of  any  pitch  is  the  number  by  which 
the  ordinate  of  the  actual  response  curve  for  the  given  pitch 
must  be  multiplied  to  give  the  ordinate  of  the  ideal  curve. 

These  correction  factors  are  obtained  from  a  correction 
curve,  c,  Fig.  126,  determined  in  the  following  manner. 
The  sixty-one  ordinates  of  the  ideal  curve  h  corresponding 
to  semitones  of  the  scale  are  divided  by  the  corresponding 
ordinates  of  the  response  curve  a ;  the  quotients  are  the 
correction  factors  for  these  particular  pitches.  These  fac- 
tors are  then  plotted  on  the  chart,  and  a  correction  curve 
is  drawn  through  the  points  as  shown  at  c.  The  correction 
curve  is  an  inverse  of  the  response  curve,  where  one  has  a 
peak  the  other  has  a  corresponding  valley. 

164 


CORRECTING  ANALYSES  OF  SOUND  WAVES 


It  is  convenient  to  make  a  correction  curve  showing  the 
factors  for  all  pitches,  since  a  number  of  photographs  of 
sounds  are  often  made  with  the  phonodeik  in  the  same  con- 
dition, and  the  analyses  of  all  are  corrected  from  the  same 
response  curve. 

The  analysis  of  a  curve  having  been  recorded  on  a  card, 
as  explained  on  page  123,  the  correction  factors,  k,  are 
measured  from  the  chart  of  the  correction  curve,  and  are 


NO.IG90 

CASE  SCHOOL  OF  APPLIED  SCIENCE                        DEPARTMENT  OF  PHY 

ANALYSIS  OF  SOUND-WAVES 

Source  ©^Q  an  T\ >0€»  —  '"9^eeA\es5  ©Vvoe" 

SICS 

DateabY\\  2.3, 

1-10 

Tone           ^    C%       '        Abs.  N  2.6  0 

Purpose  OLirtaVs^jis 

•|3I.T 

component,  n 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

na„ 

+  ZZ.L 

+  99,6 

+  lOLI 

+  7C.I 

-49.1 

+  ^.a 

t21,9 

-  n.s^ 

-  7.1 

nb„ 

-87.2 

-  IS 

-  26.1 

-  \0.(o 

-  3.6 

-66.7 

-36.9 

-  21.7 

nA„ 

94.^ 

nz.o 

7C.9, 

SI.Co 

S0.3 

45".0 

7/./ 

38,71 

22.8 

k„,(<708) 

0.(6 

0.7 

Z.I 

0.3 

l.o 

0.1 

OS 

0.6 

nA„k„ 

S7.d 

32.^ 

/Ci.l 

ICI.Z 

46.4 

27.0 

28.4 

13.^ 

/3.8 

[nA„k„p 

A„ 

96.5 

CC.O 

/9,2 

/0.3 

8.1 

6.1 

8.9 

4.3 

2.3 

P„ 

7C.' 

3/3.° 

337.' 

347' 

3^4.' 

Z90.' 

zsz.' 

zsz.' 

A„k„ 

+6.2 

5-4.7 

40.3 

9.3 

8.4 

3.9 

3.6 

2.2 

/.4 

Amplitude,^ 

20.3 

ai.o 

17.7\ 

4/ 

3.7 

1,7 

A6 

10 

0.6 

Phase 

IC7.' 

ZSO." 

4/,* 

288.' 

air 

Intensity,  % 

Remarks                                                                        Sum                 Analyzed  '^"TP.-^f 

Synthesized 

Fig.  127.    Card  form  for  the  record  of  the  analysis  of  a  sound  wave. 


recorded  on  the  line  below  the  multiple  ampHtudes  nAn, 
Fig.  127.  The  harmonic  scale  (described  on  page  169)  is 
placed  on  the  chart  with  its  first  line  on  the  pitch  of  the 
fundamental  of  the  tone  analyzed.  The  ordinates  of  the  cor- 
rection curve  opposite  the  several  harmonic  points  are  the 
correction  factors  for  the  corresponding  components ;  the 
ordinates  are  measured  with  a  millimeter  scale.  For  in- 
stance, the  fundamental  tone  had  the  pitch  260,  for  which 
the  phonodeik  responded  too  much,  as  shown  by  the  high 

165 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


peak  at  d;  for  this  point  the  correction  curve  has  an  ordinate, 
e,  equal  to  0.6,  which  is  the  factor  by  which  the  amphtude 
of  a  tone  of  pitch  260,  as  recorded  by  the  phonodeik,  must 
be  multipUed  to  reduce  it  to  the  ideal  amphtude.  For  the 
fourth  component,  at  point  4  on  the  scale,  the  response  / 
is  too  small;  the  ordinate  g  of  the  correction  curve  is  2.1, 
the  factor  by  which  the  response  for  this  particular  pitch, 
1040,  must  be  multiplied  to  make  it  equal  to  the  ideal.  A 
correction  factor  is  obtained  in  this  manner  for  each  indi- 
vidual component. 

The  products,  nAnkn,  are  found  and  each  is  divided  by  the 
order  of  its  respective  component,  1,  2,  3,  etc.,  giving  the 
corrected  amplitudes,  Ankn',  these  corrected  amplitudes 
are  presumed  to  be  proportional  to  the  actual  amplitudes 
of  the  components  of  the  original  sound  wave  in  air  before 
it  entered  the  horn  of  the  recording  apparatus.  For  com- 
parison the  corrected  amplitudes  are  expressed  as  percent- 
ages of  the  sum  of  all  the  amplitudes,  that  is,  the  sum  of  all 
the  component  amplitudes  is  equal  to  100. 

Graphical  Presentation  of  Sound  Analyses 

The  object  of  the  analysis  of  sound  waves  is  the  quanti- 
tative determination  of  the  causes  of  tone  quality.  The 
analyses  give  directly  the  amplitudes  and  phases  of  the 
various  components  of  a  sound.  The  phases  of  the  com- 
ponents which  presumably  have  little  effect  upon  the  tone 
quality  are  not  considered  at  the  present  time.  Tone 
quahty  seems  to  depend  only  upon  the  relative  intensities 
of  the  component  tones.  A  simple  comparison  of  the 
amplitudes  of  the  various  components  gives  an  inadequate 
idea  of  the  effects  perceived  by  the  ear;  in  fact,  the  rela- 
tion between  amplitude  and  loudness  varies  slightly  for 

166 


CORRECTING  ANALYSES  OF  SOUND  WAVES 


different  ears  and  for  different  frequencies.  The  relative 
energies  of  the  components,  which  may  be  derived  from  the 
observed  and  corrected  amphtudes,  while  not  corresponding 
exactly  to  the  intensities  as  perceived  by  the  ear,  afford  a 
close  approximation,  and  these  energies  will  be  used  in  the 
present  discussion  to  represent  loudness. 

If  n  is  the  order  of  a  component  tone  in  the  natural  se- 


No.  [690 

CASE  SCHOOL  OF  APPLIED  SCIENCE                        DEPARTMENT  OF  PHY 
ANALYSIS  OF  SOUND-WAVES 

Source  ©rao^rv  'Pi pe  — '"p-eed Ves3  ©boe'.' 

SICS 

1-10 

Tone           'C,          *       Abs.  N  Z60 

Purpose 

DateGLS^^5l23^ 

component,  n 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

+  zza 

+  99.6 

+  loi.i 

t  76.1 

f  i^.S 

-  ^ai 

-t- 

+ 

-  w.s 

-  71 

nb. 

+  9^.1 

-  872 

-  -^2.5 

-  7S 

-  26.1 

-  /0.6 

-  3.a 

-66.7 

-  36.9 

-  21.7 

nA„ 

96.5 

/32.0 

lo$f\ 

S\.(6 

5'0.3 

45-.0 

38.7 

2.2.8 

k„,(|70ft) 

0.6 

on 

l.s 

2.1 

0.9 

1.0 

o.^, 

0.^ 

0.5- 

nA,k, 

S7^ 

32.^ 

/6/,3 

46.1 

Z7.0 

28:^ 

/9.4 

/f.8 

26329. 

260/6. 

2/5-3. 

25*30. 

80  7. 

376. 

J30. 

A„ 

66.0 

36.5" 

19.2 

/0.3 

8.4 

6.4 

8.9 

4.3 

2^ 

P„ 

3  19.'* 

337° 

330." 

252.' 

252." 

^6.2 

^0.3 

8.i- 

3.3 

3.6 

2.2 

/4 

Amplitude,  % 

25-^ 

2.^.0 

17.7 

4.1 

3.7 

/.7 

/,6 

/.o 

0.6 

Phase 

o.° 

103' 

^or 

3;o,° 

4/." 

288.° 

Intensity,  % 

4.7 

n.s 

37.6 

3.0 

3.5" 

/.o 

l\ 

0^ 

Remarks                                                                    Sum                Analyzed  V 

Synthesized 

Fig.  128.    Card  form  for  the  record  of  the  analysis  of  a  sound  wave. 


ries,  and  AJ-Zn  its  corrected  amphtude,  then  {nAJir)-  is  a 
number  proportional  to  its  energy  or  intensity,  as  was  ex- 
plained in  Lecture  II.  As  has  been  mentioned  in  this  Lec- 
ture, the  numbers  nAJZn  are  found  in  the  process  of  deriving 
the  corrected  amplitudes ;  the  squares  of  these  numbers 
are  proportional  to  the  energies  of  the  several  components. 
The  record  is  completed  by  computing  the  percentage 
intensity  of  each  component,  Fig.  128,  that  is,  the  intensity 
of  each  component  on  the  supposition  that  the  loudness  of 
the  original  complex  sound  as  a  whole  is  represented  by  100. 

167 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


The  organ-pipe  curve  which  has  been  used  for  the  explana- 
tion of  the  method  of  analysis  contains  twelve  components, 
as  was  proved  by  its  synthetic  reproduction,  while  many 
of  the  curves  studied  have  twenty  or  more  components. 
In  the  investigation  of  several  thousand  such  curves  there 
are  hundreds  of  thousands  of  numerical  operations,  most 
of  which  are  performed  with  the  aid  of  calculating  machines, 
adding  machines,  slide  rules,  and  tables  of  squares  and  of 
products. 

The  final  results  of  the  analysis  which  are  in  the  form  of 
the  relative  intensities  of  the  several  partial  tones,  are  most 
clearly  presented  when  these  intensities  are  plotted  against 
a  logarithmic  scale  of  frequencies.  A  logarithmic  scale  of 
frequencies  is  one  in  which  the  spaces  are  proportional  to 
the  ratios  of  the  frequencies ;  it  is  represented  in  actual 
sounds  by  the  successive  tones  of  a  pianoforte.  For  instance, 
the  successive  octave  points  represent  tones  having  fre- 
quencies in  the  ratio  of  1  :  2  :  4  :  8  :  16,  etc. ;  the  ratio  is  the 
same  throughout,  and  the  actual  distance  in  inches  from  any 
note  to  its  octave  on  the  scale,  and  also  on  the  piano  key- 
board, is  the  same  whether  the  note  is  in  the  lower  part  of 
the  scale  where  the  number  of  vibrations  is  small,  or  whether 
it  is  in  the  upper  part  of  the  scale  having  several  thousand 
vibrations  per  second.  The  same  is  true  of  any  other  ratio, 
and  hence  the  distance  between  any  two  specified  partial 
tones  is  constant  throughout  the  logarithmic  scale,  and  is 
independent  of  the  pitch  of  the  fundamental. 

For  graphically  presenting  the  analyses,  a  standard  form 
of  chart  has  been  prepared,  shown  in  Fig.  129,  at  the  bottom 
of  which  is  a  logarithmic  scale  of  frequencies.  Equidistant 
vertical  lines  are  drawn  through  points  corresponding  to  the 
semitone  intervals  of  the  equally  tempered  musical  scale 

168 


CORRECTIXG  ANALYSES  OF  SOUND  WAVES 


in  International  Pitch  from  Co  =  129  to  C:  =  4138;  these 
lines  may  be  compared  to  the  strings  of  a  piano  from  one 
octave  below  middle  C  to  the  highest  note. 

In  addition  to  the  prepared  charts  a  bevel-edged  harmonic 
scale  is  provided,  with  special  rulings,  so  spaced  as  to  corre- 
spond to  the  intervals  of  the  natural  series  of  partial  tones, 
or  harmonic  tones,  from  1  to  30.    This  scale,  of  course,  will 


\ 


1  \       \  \ — \ — I   I  I  I  I  I  I  I  I  I  llllllllllllll 

2  3  4        5      6     7    8    9  10  15        20      25  30 


IS 


IT 


7    8    9  10 


FTT 

20  2 


i29     -T^rju::^     259  517  1035  2069  4138 

Fig.  129.    Methods  of  diagraming  the  analyses  of  sound  waves. 


fit  onl}'  a  particular  spacing  ;  in  the  practical  work  two  sizes 
of  charts,  and  two  sizes  of  harmonic  scales,  are  used ;  in 
the  larger  the  semitone  interval  is  one  centimeter  and  an 
octave  measures  12  centimeters,  while  in  the  other  the 
intervals  are  half  this  size. 

The  diagram  of  an  analysis  is  made  by  placing  the  har- 
monic scale  on  the  chart  so  that  the  first  line  corresponds 
to  the  actual  pitch  of  the  fundamental  of  the  original  sound ; 
the  other  divisions  of  the  scale  then  show  by  their  locations 

169 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


the  frequencies  of  the  true  harmonic  overtones  of  this 
fundamental.  An  ordinate  is  measured  through  each  har- 
monic point,  equal  in  length,  in  milhmeters,  to  the  relative 
intensity,  or  percentage  of  intensity,  of  the  corresponding 
partial.  Since  the  total  intensity  of  all  the  components  is 
100  per  cent,  the  sum  of  all  the  ordinates  on  the  diagram 
for  any  sound  is  100  millimeters.  The  upper  ends  of  the 
ordinates  are  sometimes  represented  by  circles,  the  circle 
for  the  fundamental  being  larger  than  the  others  and  having 
a  black  center. 

Fig.  129,  a,  is  the  diagram  of  the  analysis  of  the  organ- 
pipe  curve  (Fig.  96)  the  data  for  which  are  given  in  the 
bottom  line  of  the  card  shown  in  Fig.  128.  The  pitch  of 
this  tone  is  Cs  =  259,  and  the  harmonic  scale  is  placed  on 
the  chart  to  correspond.  The  loudness  of  each  partial  tone 
is  the  height  of  the  corresponding  circle  above  the  base 
line;  in  this  instance  the  third  and  fourth  partials  are  the 
loudest.  The  pitch  of  each  partial  is  shown  by  its  position 
with  respect  to  the  scale  at  the  bottom  of  the  chart.  The 
slender  black  triangles  have  no  significance  except  to  make 
the  lengths  of  the  ordinates  more  conspicuous. 

In  studying  the  analysis  of  sounds  from  certain  sources, 
it  is  helpful  to  draw  curves  through  the  upper  ends  of  the 
ordinates.  Fig.  129,  h,  is  such  a  diagram  of  the  analysis  of 
the  vowel  a  in  father,  a  photograph  of  which  is  given  in  Fig. 
160,  page  219. 

The  ordinates  of  these  diagrams  show  the  distribution  of 
the  energy  in  the  sound  with  reference  to  its  own  harmonic 
partial  tones  which  have  definite  pitches.  A  single  analysis 
gives  little  or  no  information  with  regard  to  intermediate 
pitches,  since  the  sound  analyzed  can  have  no  intensity 
whatever  for  pitches  other  than  those  of  its  own  partials. 

170 


CORRECTING  ANALYSES  OF  SOUND  WAVES 


A  second  analysis  of  a  sound  from  a  given  source,  as  in 
voice  analysis,  intoned  at  a  different  pitch  from  that  of 
the  first,  wdll  have  its  partials  at  pitches  intermediate  be- 
tween those  already  found.  By  comparing  many  analyses 
a  curve  can  be  drawn  which  shows  the  general  distribution 


I  I  I       I      I     I    1    I   I  I  I  I  1  I  I 

I  2  3  i       5      6     7    8    S  10  15 

Fig.  130.    Distribution  of  energy  in  sounds  from  various  sources. 

of  energ\^  from  the  source.  The  usefulness  of  such  curves 
in  connection  the  study  of  vowel  tones  is  more  fully 

explained  on  pages  220  and  228. 

These  diagrams  and  curves  sho^^dng  the  distribution  of 
the  energy  in  a  sound,  are  not  unlike  the  spectrum  charts 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


and  emission  curves  obtained  in  the  study  of  light  sources. 
Corresponding  to  a  monochromatic  hght  we  have  what 
may  be  called  the  ''mono-pitched"  sounds  of  the  tuning 
fork.  This  sound  is  simple,  containing  but  a  single  com- 
ponent, and  its  diagram  consists  of  one  ''strong  line,"  as 
shown  in  Fig.  130.  Other  sources  emit  complex  sounds, 
the  energy  being  variously  distributed  among  the  several 
partials,  as  shown,  for  particular  instances,  in  the  figure. 
The  horn  gives  the  most  uniformly  distributed  emission  of 
sound  energy,  and  its  tone  may  be  said  to  correspond  acous- 
tically to  white  light.  The  characteristics  of  instrumental  and 
vocal  tones  are  more  fully  discussed  in  the  succeeding  lectures. 

Verification  of  the  Method  of  Correction 

As  a  test  of  the  sufficiency  of  the  method  which  has  been 
developed  for  correcting  analyses,  one  hundred  and  thirty 
photographs  of  tones  from  nine  different  instruments  were 
made  with  four  distinctly  different  combinations  of  horn 
and  diaphragm,  giving  for  each  tone  four  sets  of  curves 
which  are  wholly  unlike.  A  long  horn  was  used  with  a  large 
and  a  small  diaphragm,  and  also  a  short  horn  with  each 
diaphragm  ;  the  responses  were  such  that  the  peaks  in  one 
instance  corresponded  in  pitch  with  the  valleys  in  another. 
Response  and  correction  curves  were  made  for  each  combi- 
nation. After  correction  the  various  analyses  of  any  one 
tone  were  identical. 

Fig.  131  shows  the  photographs  of  the  tone  from  an  organ 
pipe  made  with  three  horn-and-diaphragm  combinations ; 
these  would  hardly  be  taken  for  records  of  the  same  sound. 
After  analysis,  the  components  for  each  curve  were  cor- 
rected for  horn  and  diaphragm  effects  and  then  recom- 
pounded  with  the  synthesizer,  the  three  corrected  curves 

172 


CORRECTING  ANALYSES  OF  SOUND  WAVES 


Fig.  131.    Three  curves  for  the  same  tone,  made  under  different  conditions. 


being  shown  in  Fig.  132.  These  curves  show  how  success- 
ful the  method  is  in  reducing  unhke  curves  for  the  same  sound 
to  practical  identity. 


Fig.  132.    The  three  cur\'es  of  Fig.  131  corrected  for  instrumental  effects. 

173 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Quantitative  Analysis  of  Tone  Quality 

In  Lectures  III,  IV,  and  V  there  has  been  described  a 
quantitative  method  for  the  analytical  study  of  tone  quality. 
The  method  includes  the  arrangement  of  the  working  appara- 
tus, and  schemes  for  computing,  reducing,  presenting,  and 
filing  the  results.  The  results  so  obtained  are  expressed  in 
terms  of  the  relative  loudness  of  the  various  partial  tones 
of  an  instrument.  A  determination  of  the  relative  loud- 
ness of  the  sounds  of  one  instrument  as  compared  with 
another  is  no  doubt  of  much  interest,  but  this  is  not  included 
in  the  present  discussion. 

In  the  definitive  study  of  a  musical  instrument  or  voice  it 
is  desirable  that  a  large  number  of  tones  be  photographed, 
perhaps  four  per  octave,  three  semitones  apart,  through- 
out the  whole  compass ;  the  tones  should  be  sounded  in 
three  different  intensities  as  m/,  and  /,  or  five  intensities 
may  be  studied,  by  adding  and  ff;  to  eliixdnate  errors, 
two  or  more  combinations  of  horn  and  diaphragm  may  be 
used ;  the  source  of  sound  may  be  placed  at  various  dis- 
tances from  the  horn ;  response  and  correction  curves  must 
be  taken  before  and  after  each  change  in  the  recording 
apparatus ;  such  a  study  of  one  instrument  may  require 
a  year's  time  for  its  completion.  Any  scheme  less  com- 
prehensive than  this  will  not  give  an  adequate  idea  of  the 
tone  quality  of  a  musical  instrument. 


174 


LECTURE  VI 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 
Generators  and  Resonators 

Previous  lectures  have  demonstrated  that,  in  general, 
the  sounds  from  musical  instruments  are  composite,  that 
is,  all  those  which  can  be  said  to  have  characteristic  quality- 
are  made  up  of  a  larger  or  smaller  number  of  partial  tones 
of  various  degrees  of  loudness.  A  scientific  definition  of 
the  quality  of  a  musical  tone  requires  a  statement  of  what 
particular  partial  tones  enter  into  its  composition  and  of 
the  intensities  and  phase  relations  of  these  partials.  In 
order  to  understand  a  musical  instrument,  we  need  to 
know  how  its  tones  are  generated  and  controlled  by  the 
performer. 

The  sound  producing  parts  of  a  musical  instrument,  in 
general,  perform  two  distinct  functions.  Certain  parts  are 
designed  for  the  production  of  musical  vibrations.  The 
vibrations  in  their  original  form  may  be  almost  inaudible, 
though  \4gorous,  because  they  do  not  set  up  waves  in  the 
air,  as  is  illustrated  by  the  \ibrations  of  the  string  of  a 
violin  without  the  body  of  the  instrument ;  or  the  vibra- 
tions may  produce  a  very  undesirable  tone  quality  because 
they  are  not  properly  controlled,  as  in  the  case  of  the  reed 
of  a  clarinet  ^dthout  the  body  tube.  Other  parts  of  the 
instrument  receive  these  vibrations,  and  by  operation  on  a 

175 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


larger  quantity  of  air  and  by  selective  control,  cause  the 
instrument  to  send  out  into  the  air  the  sounds  which  we 
ordinarily  hear.  These  parts,  which  may  be  referred  to  as 
generator  and  resonator,  are  illustrated  by  the  following 
combinations :  a  tuning-fork  generator  and  its  box  resona- 
tor ;  the  strings  and  soundboard  of  a  piano ;  the  reed  and 
body  tube  of  a  clarinet ;  the  mouth  and  body  tube  of  an 
organ  pipe ;  the  vocal  cords  and  mouth  cavities  of  the 
voice.  In  the  piano  the  soundboard  acts  as  a  universal 
resonator  for  all  the  tones  emitted  by  the  instrument ;  in 
the  organ  each  pipe  constitutes  its  own  separate  resonator ; 
in  the  flute  the  body  tube  is  adjusted  to  various  different 
conditions  by  means  of  holes  and  keys,  each  condition  serv- 
ing for  several  tones. 

The  resonator  cannot  give  out  any  tones  except  those 
received  from  the  generator,  and  it  may  not  give  out  all  of 
these.  The  generator  must  therefore  be  capable  of  pro- 
ducing the  components  w^hich  we  wish  to  hear,  and  these 
must  in  turn  be  emitted  in  the  desired  proportion  by  the 
resonator.  If  the  generator  produces  partial  tones  which 
are  undesirable,  the  resonator  should  be  designed  so  that 
it  will  not  reproduce  them ;  if  the  generator  produces  tones 
which  are  of  musical  value  but  which  the  resonator  does 
not  reproduce,  we  do  not  hear  them,  and  it  is  as  though  they 
were  not  produced  at  all.  It  follows  that  we  can  hear  from 
a  given  instrument  nothing  except  what  is  produced  by  the 
generator,  and  further  we  can  hear  nothing  except  what  is 
also  reproduced  by  the  resonator ;  hence  it  may  be  that  the 
most  important  part  of  an  instrument  is  its  resonator.  The 
quality  of  an^^  tone  depends  largely  upon  the  kind  and  degree 
of  sympathy,  or  resonance,  which  exists  between  the  genera- 
tor and  the  resonator. 

176 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


Resonance 

Every  vibrating  body  has  one  or  more  natural  periods  in 
which  it  \dbrates  easily  ;  to  tune  a  sounding  body  is  to  adjust 
its  natural  period  to  a  specified  frequency.  If  a  body  capa- 
ble of  vibration  is  excited  by  any  means  whatever,  and  the 
exciting  cause  is  removed,  the  body  will  usually  vibrate  freely 
in  its  natural  frequency,  or  with  its  free  period.  If  the 
exciting  cause  operates  in  this  same  frequency,  the  two  are 
in  resonance,  that  is,  they  are  in  tune ;  under  these  condi- 
tions the  response  of  the  body  receiving  the  vibration  is  a 
maximum.  If  the  exciting  cause  differs  in  frequency  but 
slightly  from  that  natural  to  the  other  body,  there  will  still 
be  response  but  in  a  lesser  degree,  that  is,  the  resonance  is 
not  so  sharp.^^  When  the  two  bodies  are  quite  out  of  tune, 
there  will  be  very  little  resonance,  and  while  the  second 
body  may  still  be  made  to  vibrate,  the  response  will  be 
small.  These  conditions  are  well  illustrated  by  the  response 
curves  described  in  the  previous  lecture. 

When  the  resonator  is  out  of  tune  with  the  generator,  it 
is  often  made  to  vibrate  with  the  generator,  and  it  is  then 
said  to  have  forced  vibration.  In  forced  vibration,  the  two 
bodies  have  different  natural  frequencies,  and  the  resulting 
forced  frequency  is  in  general  not  that  natural  to  either 
body,  each  drawing  the  other  more  or  less  to  a  common 
intermediate  frequency.  In  musical  instruments  usualh^ 
the  generator  is  much  less  influenced  than  is  the  resonator  ; 
for  instance,  a  tuning  fork  in  connection  with  a  resonance 
box  not  exactly  in  tune,  draws  the  air  in  the  box  to  its  own 
frequency  much  more  easily  than  the  air  draws  the  fork. 
Koenig  found  that  for  a  fork  of  256  vibrations  per  second 
the  maximum  alteration  of  its  frequency  due  to  the  draw- 
N  177 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


ing  effect  of  the  resonance  box  is  produced  when  the  box 
has  a  natural  frequency  of  either  248  or  264.  The  fork 
in  causing  forced  vibration  of  the  air  in  the  box  draws  the 
air  from  a  frequency  of  248  to  256.036  in  the  first  case,  and 
from  264  to  255.964  in  the  second ;  that  is,  the  box  being 
out  of  tune  by  8  vibrations,  the  fork  is  forced  out  of  its 
natural  frequency  by  0.036  vibration.  When  the  box  is 
out  of  tune  by  a  musical  semitone,  the  effect  on  the  fork 
is  less,  being  about  0.025  vibration.^^ 

When  a  body  is  exactly  in  tune  with  the  generator,  that 
is,  when  it  is  in  resonance,  it  may  take  up  the  vibrations 
with  great  ease  and  vigor ;  such  a  response  is  often  called 
sympathetic  vibration.  This  is  often  disagreeably  illustrated 
by  the  rattle  of  bric-a-brac  in  a  music  room,  or  by  the  buzz 
of  some  part  of  the  action  of  a  piano  or  of  a  machine.  Sym- 
pathetic vibration  is  demonstrated  by  means  of  two  forks 
which  are  exactly  in  tune ;  if  one  fork  is  sounded  loudly 
for  a  few  seconds,  the  other  fork  is  set  in  audible  vibration, 
the  only  medium  of  communication  being  the  air. 

There  are  two  distinct  kinds  of  resonators.  One  kind 
having  no  definite  vibration  frequency  of  its  own,  responds 
to  tones  of  any  frequency  and  to  combinations  of  these ; 
it  can  reproduce  all  gradations  of  tone  quality.  A  plate, 
such  as  the  soundboard  of  a  piano,  is  representative  of  this 
kind  of  resonator.  The  second  type  of  resonator  possesses 
a  more  or  less  definite  natural  frequency  and,  because  of 
selective  control,  it  reproduces  sounds  of  particular  quality 
only.  Such  a  resonator  will  respond  not  only  to  tones 
corresponding  to  its  fundamental,  but  also  to  tones  in  uni- 
son wdth  its  overtones.  The  second  kind  of  resonator  is 
typified  by  the  cylindrical  brass  box  of  the  standard  tun- 
ing fork  described  on  page  51.    This  box  has  a  very  definite 

178 


TONE  QUALITIES  OE  MUSICAL  INSTRUMENTS 


fundamental  frequency  and  overtones  which  are  high  in 
pitch  and  not  in  tune  with  any  overtone  of  the  fork ;  there- 
fore only  the  fundamental  of  the  fork  is  reinforced,  and  the 
result  is  a  pure  simple  tone. 

A  resonator  does  not  create  any  sound ;  it  can  only  take 
up  the  energy  of  vibration  of  the  generator  and  give  it  out 
in  a  different  loudness.  It  follows  that  for  a  given  blow  to 
a  fork  or  a  string,  the  more  perfect  the  tuning  of  the  resona- 
tor, the  louder  will  be  the  sound  and  the  shorter  will  be  its 
duration.  If  the  strings  of  two  different  pianos  are  struck 
with  the  same  force  of  blow,  that  piano  which  gives  the 
loudest  sound  will  probably  have  the  shortest  duration  of 
tone,  while  the  one  which  begins  the  sound  with  moderate 
loudness  will  continue  to  sound  longer  or  will  ''sing"  better. 

The  loudness  and  the  duration  ot  the  sound  from  an  instru- 
ment are  dependent  upon  the  damping  or  absorption  of 
the  vibration  in  the  instrument  and  its  surroundings.  The 
energy  of  the  waves  which  travel  outward  from  a  sound- 
ing body  is  derived  from  the  vibration  of  the  body ;  usually 
not  all  of  the  energy  of  vibration  is  transferred,  some  being 
absorbed  and  transformed  into  heat  through  friction  and 
the  viscosity  of  the  body.  When  the  loss  of  energy  is  rapid, 
the  amplitude  of  vibration  decreases  rapidly,  and  the  vibra- 
tions are  said  to  be  damped These  effects  must  be  con- 
sidered with  resonance  and  consonance  in  the  complete 
study  of  musical  instruments. 

Effects  of  Material  on  Sound  Waves 

Both  the  tones  generated  by  a  musical  instrument  and 
those  reproduced,  as  well  as  those  absorbed  or  damped,  de- 
pend in  a  considerable  degree  upon  the  material  of  which 
the  various  parts  of  the  instrument  are  constructed.  While 

179 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


this  fact  is  well  known  and  commonly  made  use  of  in  connec- 
tion with  certain  classes  of  instruments,  its  truthfulness  is 
often  denied  by  the  devotees  of  other  instruments.  The 
question  of  the  influence  of  the  material  of  which  the  body 
tube  of  a  flute  is  made  has  not  been  settled  after  more  than 
seventy  years  of  widespread  discussion.      How  does  the 

tone  from  a  gold  or 
silver  flute  differ  from 
that  of  a  wooden  flute  ? 
It  was  this  specific  ques- 
tion that  suggested  the 
investigations  which, 
having  passed  much  be- 
yond the  original  in- 
quiry, have  furnished 
the  material  upon 
which  this  course  of  lec- 
tures is  based. 

The  following  experi- 
ments, suggested  by 
those  of  Schafhautl 
(Munich,  1879), indi- 
cate the  great  changes 
in  the  tone  of  an  organ 
pipe  which  may  be  pro- 
duced by  effects  pass- 
ing through  the  walls.  Three  organ  pipes  are  provided, 
as  shown  in  Fig.  133.  The  first  pipe,  of  the  ordinary 
type  used  in  physical  experiments,  is  made  of  wood  and 
sounds  the  tone  G2  =  192.  Two  pipes  having  exactly  the 
same  internal  dimensions  as  the  wooden  one  are  made 
of  sheet  zinc  about  0.5  millimeter  thick.    One  of  the  zinc 

180 


Fig.  133.    Organ  pipes  for  demonstrating  the 
influence  of  the  walls  on  the  tone. 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


pipes  has  been  placed  inside  a  zinc  casing  to  form  a  double- 
walled  pipe,  with  spaces  two  centimeters  wide  between  the 
walls ;  the  outer  wall  is  attached  to  the  inner  one  only  at 
the  extreme  bottom  on  three  sides,  and  just  above  the 
upper  lip-plate  on  the  front  side.  These  two  pipes  have 
exactly  the  same  pitch,  giving  a  tone  a  little  flatter  than 
F2,  w^hich  is  more  than  two  musical  semitones  lower  than 
that  of  the  wooden  pipe  of  the  same  dimensions. 

Using  the  single-walled  zinc  pipe  one  can  produce  the 
remarkable  effect  of  choking  the  pipe  till  it  actually  squeals. 
When  the  pipe  is  blown  in  the  ordinary  manner,  its  sound 
has  the  usual  tone  quality.  If  the  pipe  is  firmly  grasped 
in  both  hands  just  above  the  mouth,  it  speaks  a  mixture 
of  three  clearly  distinguished  inharmonic  partial  tones, 
the  ratios  of  which  are  approximately  1 :  2.06 :  2.66.  The 
resulting  unmusical  sound  is  so  unexpected  that  it  is  almost 
startling,  the  tone  quality  ha\dng  changed  from  that  of  a 
flute  to  that  of  a  tin  horn. 

Experiments  with  the  double-walled  pipe  are  perhaps 
more  convincing.  While  the  pipe  is  sounding  continuously, 
the  space  between  the  walls  is  slow^ly  filled  with  water  at 
room  temperature.  The  pipe,  with  the  dimensions  of  a 
wooden  pipe  giving  the  tone  G2,  w^hen  empty  has  the 
pitch  F2,  and  when  the  walls  are  filled  with  water  the 
pitch  is  E2;  during  the  filling  the  pitch  varies  more  than 
a  semitone,  first  rising  then  falling.  While  the  space  is 
filling,  the  tone  quality  changes  conspicuously  thirty  or 
forty  times. 

After  the  demonstration  of  these  effects,  one  mil  surely 
admit  that  the  quality  of  a  wind-instrument  may  be  affected 
by  the  material  of  its  body  tube  to  the  comparatively  small 
extent  claimed  by  the  player.    The  flute  is  perhaps  espe- 

181 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


cially  susceptible,  to  this  influence  because  its  metal  tube 
is  usually  only  0.3  millimeter  thick.  It  is  conceivable  that 
the  presence  or  absence  of  a  ferrule  or  of  a  support  for  a  key 
might  cause  the  appearance  or  disappearance  of  a  partial 
tone,  or  put  a  harmonic  partial  slightly  out  of  tune. 

The  traditional  influence  of  different  metals  on  the  flute 
tone  are  consistent  with  the  experimental  results  obtained 
from  the  organ  pipe.  Brass  and  German  silver  are  usually 
hard,  stiff,  and  thick,  and  have  but  little  influence  upon 
the  air  column,  and  the  tone  is  said  to  be  hard  and  trumpet- 
like. Silver  is  denser  and  softer,  and  adds  to  the  mellow- 
ness of  the  tone.  The  much  greater  softness  and  density 
of  gold  adds  still  more  to  the  soft  massiveness  of  the  walls, 
giving  an  effect  like  the  organ  pipe  surrounded  with  water. 
Elaborate  analyses  of  the  tones  from  flutes  of  wood,  glass, 
silver,  and  gold  prove  that  the  tone  from  the  gold  flute 
is  mellower  and  richer,  having  a  longer  and  louder  series 
of  partials,  than  flutes  of  other  materials. 

Mere  massiveness  of  the  walls  does  not  fulfill  the  desired 
condition ;  a  heavy  tube,  obtained  from  thick  walls  of 
brass,  has  such  increased  rigidity  as  to  produce  an  undesir- 
able result ;  the  walls  must  be  thin,  soft,  and  flexible,  and 
must  be  made  massive  by  increasing  the  density  of  the  ma- 
terial. The  gold  flute  tube  and  the  organ  pipe  surrounded 
with  water,  are,  no  doubt,  similar  to  the  long  strings  of  the 
pianoforte,  which  have  a  rich  quality ;  these  strings  are 
wound  or  loaded,  making  them  massive,  while  the  flexi- 
bility or  softness"  is  unimpaired.  The  organ  pipe  partly 
filled  with  water  is  like  a  string  unequally  loaded,  its  partials 
are  out  of  tune  and  produce  a  grotesque  tone.  A  flute  tube 
having  no  tone  holes  or  keys  is  influenced  by  the  manner 
of  holding ;    certain  overtones  are  sometimes  difficult  to 

182 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


produce  until  the  points  of  support  of  the  tube  in  the  hands 
have  been  altered. 

Beat-Tones 

When  two  simple  tones  are  sounding  simultaneously,  in 
general,  beats  are  produced,  equal  in  number  to  the  differ- 
ence of  the  frequencies.  AATien  the  beats  are  few  per 
second,  the  separate  pulsations  are  easily  detected.  "\Mien 
the  beats  are  many,  the  ear  does  not  perceive  the  sep- 
arate  pulses,  and  instead   the   sensation   is   that   of  a 


a  I 

C     .  4  *  • 


Fig.  134.    Photograph  of  beats  produced  by  two  tuning  forks,  giving  the  effect 
of  a  third  tone,  called  a  beat-tone. 

third  tone,  which  is  as  distinct  and  as  musical  as  the  two 
generating  tones,  and  which  has  a  frequency  equal  to 
the  difference  in  the  frequencies  of  the  two  generators ; 
that  is,  its  frequency  is  equal  to  the  number  of  beats  if 
such  rapid  beats  could  be  heard.  This  tone  is  called  a 
heat-to72e 

Fig.  134  is  a  photograph  of  the  weaves  from  two  tuning 
forks  ha^1ng  frequencies  of  Ce  =  2048  and  De  =  2304, 
respectively,  which  are  in  the  ratio  of  8:9.  If  the  two 
sets  of  waves  are  in  like  phase  at  a  certain  point,  they 
combine  and  produce  a  curve  of  large  amplitude,  as  at  a, 
signifying  a  loud  sound.  This  condition  is  repeated  at 
regular  intervals  along  the  wave  train,  as  at  h,  which  is 

183 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


exactly  8  waves  of  one  tone  and  9  waves  of  the  other 
from  a. 

A  point  c,  midway  between  a  and  b,  is  4  waves  of  one  tone 
and  4 J  waves  of  the  other  from  a.  When  the  motion  at 
this  point  due  to  one  wave  is  upward,  that  due  to  the  other 
is  downward,  and  the  two  neutrahze  each  other,  producing 
the  effect  shown  in  the  curve  and  which  corresponds  to  ^ 
minimum  of  sound.  This  neutrahzing  effect  occurs  regu- 
larly between  each  two  reinforcements.  The  resultant 
sound  of  the  two  forks  waxes  and  wanes  as  does  the  out- 
line of  the  curve.  When  the  number  of  fluctuations  is  less 
than  16  per  second  the  ear  hears  the  separate  pulsations 
as  beats ;  when  the  number  of  pulsations  is  large,  the  effect 
upon  the  ear  is  that  of  a  continuous  simple  tone  of  a  fre- 
quency equal  to  the  number  of  beats  per  second ;  this  effect 
is  the  beat-tone.  There  are  256  beats  per  second  in  the 
instance  described,  and  the  ear  hears  not  only  the  two  real 
fork-tones,  Ce  =  2048  and  De  =  2304,  but  also  a  third 
beat-tone,  of  the  pitch  C3  =  256.  The  latter  sounds  just 
as  real  as  the  other  two  tones,  but  it  has  no  physical  exist- 
ence as  a  tone ;  there  is  no  vibrating  component  of  motion 
corresponding  to  the  beat-tone,  an  analysis  of  the  wave 
form  showing  only  the  two  components  due  to  the  forks. 
While  beat-tones  are  purely  subjective,  yet  they  affect  the 
ear  as  do  real  tones.  These  subjective  partials  have  great 
influence  on  the  tone  quality  of  many  instrumental  and 
vocal  sounds  as  perceived  by  the  ear.  This  influence  has 
never  been  fully  appreciated. 

Identification  of  Instrumental  Tones 

There  are  many  who,  listening  to  a  full  orchestra,  are 
able  to  distinguish  the  tones  of  a  single  instrument  even 

184 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


when  all  the  instruments  are  being  played.  We  usually 
think  this  possibihty  is  dependent  on  the  characteristic 
quality  of  the  instrument,  but  investigation  indicates  that 
tone  quality  is  only  one  of  several  perhaps  equally  important 
factors  of  identification.  Other  aids  in  the  differentiation 
are  the  attendant  and  characteristic  noises  of  the  instru- 
ment, such  as  the  scratching  of  the  bow,  the  hissing  ol  the 
breath,  and  the  snapping  of  the  plucked  strings.  A  further 
very  important  help  to  the  observer,  especiall}^  if  he  is  not  a 
trained  musician,  is  the  casual  observation  of  the  motions 
of  the  performer ;  the  synchronism  of  these  movements 
with  the  changes  of  the  melody  calls  attention  to  the  par- 
ticular instrument.  Every  one  attending  a  concert  desires 
to  see  the  musicians  as  well  as  to  hear  them;  a  seat  in  a 
concert  hall  which  allows  no  \'iew  of  the  musicians  is  con- 
sidered most  undesirable. 

It  is  very  difficult  to  keep  the  many  instruments  of  an 
orchestra  in  perfect  tune ;  indeed  it  is  almost  certain  that 
perfect  tuning  is  unattainable.  The  imperfections  of  tun- 
ing prevent  the  harmonious  blending  of  the  sounds  of  the 
various  instruments,  and  an  individual  instrument  may  be 
separated  from  the  mass  of  sound  by  its  particular  pitch, 
which  condition  will  help  to  differentiate  it  and  assist  the 
hearer  in  the  identification.  The  hearer  may  not  be  con- 
scious of  the  lack  of  tuning,  and,  indeed,  many  persons  are 
not  over-critical  in  this  respect. 

Helmholtz  says^^  that  the  proper  musical  qualities  of  the 
tone  from  a  fork  and  of  that  produced  by  blowing  across 
the  mouth  of  a  bottle,  both  being  simple,  are  identical. 
Certain  tones  of  the  flute  are  also  simple,  and  therefore  of 
the  same  quality  as  those  of  the  tuning  fork.  Experi- 
mental demonstration  has  proved  that  when  the  auditor 

185 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


is  removed  from  the  sounding  bodies  so  that  he  is  unable 
either  to  see  them  or  to  hear  the  attendant  noises,  he  can- 
not tell  whether  it  is  the  fork,  the  bottle,  or  the  flute,  that 
produces  the  tone.  It  is  a  fact  that  certain  tones  can  be 
produced  on  the  flute,  in  the  lower  register,  which  cannot 
be  distinguished  by  the  trained  musician  from  certain  tones 
of  the  \4olin,  and  like  similarities  are  possible  with  pairs 
of  other  instruments.  Of  course,  any  instrument  can  prob- 
ably produce  certain  peculiar  tones  which  are  impossible  of 
imitation  by  any  other  instrument. 

The  musician  and  the  scientist  are  interested  in  the  dis- 
tinguishing features  of  the  tone  qualities  of  the  various 
orchestral  instruments  and  of  other  sources  of  musical 
sounds.  The  true  characteristic  tone  of  an  instrument  is 
the  sustained  and  continuable  sound  produced  after  the 
sound  has  been  started  and  has  reached  what  may  be  called 
the  steady  state ;  this  steady  sound  is  usually  free  from  the 
noises  of  generation.  Systematic  analyses  covering  the 
entire  scale  in  various  degrees  of  loudness  have  been  made 
for  the  flute,  violin,  horn,  and  voice,  and  less  complete 
analyses  have  been  made  for  other  instruments ;  these 
studies  are  to  be  continued  until  a  general  survey  has  been 
made  of  the  entire  tonal  facilities  of  instrumental  music. 
The  analyses  which  have  been  completed  make  it  possible 
to  describe  the  distinguishing  characteristics  of  the  tones 
of  the  several  instruments. 

The  Tuning  Fork 

The  musical  instrument  which  gives  the  simplest  and 
purest  tone,  as  mentioned  in  Lecture  II,  is  a  tuning  fork 
in  connection  with  a  resonator.  A  photograph  of  the  tone 
from  a  fork  sounding  middle  C  =  256  is  shown  in  Fig.  135 ; 

186 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 

such  a  tone  needs  no  other  description  than  the  statement 
that  it  is  simple.  This  wave  form  will  be  recognized  as 
that  produced  by  the  simplest  possible  vibrator}^  motion, 


Fig.  135.    Photograph  of  the  simple  tone  from  a  tuning  fork. 


simple  harmonic  motion.  For  comparison  the  analysis 
of  such  a  simple  tone  is  shown  on  the  diagram  of  various 
tones  in  Fig.  130,  page  171  ;  since  the  tone  has  but  one  com- 
ponent the  diagram  consists  of  one  line  only. 


A 

h 

A  1^ 

A 

K 

N 

r. 

\ 

J 

s 

\  A 

VA 

•  /  ■ 

V 

A' 

•J  i 

V 

»■ 

a 

h 

Fig.  136.    Photograph  of  the  clang-tone  from  a  tuning  fork. 


If  a  fork  is  struck  a  sharp  blow  with  a  wooden  mallet  or 
other  hard  body,  it  can  be  made  to  give  a  ringing  sound  in 
which  the  ear  easily  distinguishes  a  high-pitched  clang-tone 

187 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


in  addition  to  the  fundamental  heard  when  the  fork  is 
sounded  with  a  soft  hammer ;  this  clang-tone  is  the  first 
natural  overtone  of  the  fork.  Fig.  136  is  a  photograph  of 
the  tone  from  a  fork  struck  with  a  wooden  mallet,  the 
kinks  in  the  wave  form  being  due  to  the  overtone  thus  pro- 
duced. Inspection  shows  that  the  relation  of  the  small  wave 
to  the  large  one  occurring  at  the  point  a  does  not  recur  till 


Fig.  137.    Photograph  of  the  tone  of  a  tuning  fork  having  the  octave  overtone. 

the  fourth  succeeding  wave,  at  6;  in  the  four  large  waves 
there  are  twenty-five  kinks  due  to  the  small  one,  that  is,  the 
frequency  of  the  overtone  is  about  6.25  times  that  of  the 
fundamental.  Since  there  is  not  an  integral  number  of  the 
smaller  waves  to  one  of  the  larger,  the  partial  is  inharmonic 
or  out  of  tune,  and  hence  the  sound  is  clanging  or  metallic 
rather  than  musical. 

When  a  tuning  fork  mounted  on  a  resonance  box  is  sounded 
by  vigorous  bowing,  it  sometimes  produces  a  strong  octave 
overtone  ;  such  a  tone  is  not  natural  to  either  the  fork  or  the 
box,  and  is  probably  due  to  some  peculiar  condition  of  the 
combination  which  has  not  yet  been  fully  explained.  A 

188 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


photograph  of  this  unusual  tone  from  a  fork  is  shown  in 
Fig.  137. 

Tuning  forks  have  been  used  as  musical  instruments  in 
connection  with  keyboards  hke  those  of  the  piano  or  organ. 
The  tones  are  remarkably  sweet  and  of  greater  purity 
than  those  obtainable  from  any  other  instrument ;  but 
the  very  fact  of  purity,  that  is,  the  absence  of  higher 
partial  tones,  renders  the  music  monotonous  and  uninter- 
esting, and  such  devices  have  not  survived  the  experimental 
stages. 

The  Choralcelo,  an  instrument  of  recent  design,"^  produces 
a  sustained  tone,  having  the  same  general  characteristics  as 
that  of  the  tuning  fork.  The  vibrations  are  produced  by 
electromagnets,  through  which  flow  interrupted,  direct,  elec- 
tric currents,  the  pulsations  of  which  are  of  the  same  periods 
as  those  of  the  bodies  to  be  set  in  motion.  The  sources  of 
sound  may  be  piano  strings  or  ribbons  of  steel  drawn  over 
a  soundboard,  which  are  set  in  vibration  by  the  direct  action 
of  the  magnets.  In  other  instances,  bars  of  wood,  alumi- 
num, or  steel  are  used  in  connection  with  resonators,  or  dia- 
phragms of  special  construction  are  fastened  to  the  ends  of 
resonant  tubes ;  soft  iron  armatures  are  attached  to  the  bars 
and  diaphragms,  which  are  set  in  vibration  by  the  pulsations 
of  the  magnets,  and  thus  the  air  in  the  resonators  is  moved 
and  the  tones  are  produced.  The  tones  so  obtained  are 
nearly  simple  in  quality,  consisting  mainly  of  a  fundamen- 
tal. The  overtones,  naturally  absent,  are  provided  by 
sounding  corresponding  generators  in  accordance  with  a 
scheme  of  tone  combinations  which  can  be  carried  out  con- 
veniently by  means  of  stops  or  controllers  operating  switches 
in  the  electrical  apparatus.  The  Choralcelo  produces  tones 
which  are  very  clear  and  vibrant  and  of  great  carrying 

189 


THE  s(  ip:xce  of  musical  sounds 


power,  due,  perhaps,  to  the 
strong  fundamental  component. 
The  combinations  of  such 
sounds  produce  unique  tonal 
effects,  of  remarkable  musical 
quality,  and  the  possibilities 
of  synthetic  tone  development 
are  great. 

The  Flute 

The  flute  in  principle  is  of 
the  utmost  simplicity ;  it  con- 
sists of  a  cylindrical  air  colimin 
a  few  inches  in  length,  set  into 
longitudinal  vibration  by  blow- 
ing across  a  hole  near  the  end 
of  the  tube  which  incloses  the 
air  column.  The  holes  in  the 
body  of  the  flute,  with  the  keys 
and  mechanism,  Fig.  138,  serve 
only  to  control  the  effective 
length  of  the  \dbrating  air 
column. 

While  the  flute  is  simple 
acoustically,  the  manipulation 
of  the  instrument  in  accordance 
with  the  requirements  of  music 
of  the  present  time,  requires  a 
key-mechanism  of  considerable 
complexity  and  of  the  finest 
workmanship.  The  flute  has 
been  developed  to  an  acous- 


.  138.    The  flute.  190 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 

tical  and  mechanical  perfection  perhaps  not  attained  by 
any  other  orchestral  instrument.  This  is  largely  due  to  the 
artistic  and  scientific  studies  of  the  instrument  made  by 
Theobald  Boehm,  of  Munich,  who  devised  the  modern 
system  of  fingering  in  1832,  and  invented  the  cyUndrical- 
bore,  metal  tube,  with  large  covered  finger  holes,  in  1847."^^ 
The  flute  gives  the  simplest  sound  of  any  orchestral  in- 
strument, and  this  is  especially  true  when  it  is  played  softly. 
The  paucity  of  overtones  causes  its  sound  to  blend  more 
readily  with  that  of  other  instruments  or  the  voice,  and 
prevents  the  poignant  expressiveness  of  the  stringed  and 
reed  instruments ;  nevertheless,  the  flute  has  an  expression 
peculiar  to  itself,  and  an  aptitude  for  rendering  certain 
sentiments  not  possessed  by  any  other  instrument.  Berlioz 
says:  ^'If  it  were  required  to  give  a  sad  air  an  accent  of 
desolation  and  of  humility  and  resignation  at  the  same  time, 
the  feeble  sounds  of  the  flute's  medium  register  would 
certainly  produce  the  desired  effect."  The  flute,  because 
of  its  agility  and  ability  to  play  detached  and  extended 
passages,  arpeggios,  and  iterated  notes,  as  well  as  because 
of  its  light  tone  quality,  is  suited  to  music  of  the  gayest 
character.  The  flute  tone  is  often  described  as  sweet  and 
tender ;  Sidney  Lanier,  himself  an  accomplished  flutist, 
describes  this  tone  in  ^^The  Symphony"  : 

'^But  presently 
A  velvet  flute-note  fell  down  pleasantly 
Upon  the  bosom  of  that  harmony, 
And  sailed  and  sailed  incessantly, 
As  if  a  petal  from  a  wild-rose  blown 
Had  fluttered  down  upon  that  pool  of  tone 
And  boatwise  dropped  o'  the  convex  side 
And  floated  down  the  glassy  tide 
And  clarified  and  glorified 
The  solemn  spaces  where  the  shadows  bide." 
191 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Sound  waves  from  the  flute  are  shown  in  Fig.  139,  in 
which  are  three  curves  for  the  same  tone,  G3  =  388,  played 

vif,  and  / ;  when  played  softly  the  tone  is  nearly  simple, 
an  increase  in  loudness  adds  first  the  octave,  and  then  still 
higher  partials. 

About  a  thousand  photographs  of  fl.ute  tones  have  been 
analyzed,  including  every  note  in  the  scale  of  the  instru- 
ment, each  in  several  degrees  of  loudness ;  flutes  made  of 
various  materials  have  been  studied,  as  wood,  silver,  gold. 


Fig.  139.    Three  photographs  of  the  tone  of  a  flute,  played  p,  mf,  and  /. 


and  glass,  and  the  effects  of  different  sized  holes  have  been 
investigated.  Some  of  the  results  of  the  analyses  of  the 
tones  of  the  gold  flute  will  be  described ;  flutes  of  other 
materials  have  the  same  general  characteristics,  except 
that  the  overtones  are  fewer  and  weaker. 

The  average  composition  of  all  the  tones  of  the  low 
register  of  the  flute,  one  octave  in  range,  when  played 
pianissimo,  is  shown  by  the  lower  line  of  Fig.  140 ;  these 
tones  are  nearly  simple,  containing  about  95  per  cent  of 
fundamental,  with  a  very  weak  octave  and  just  a  trace 

192 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


of  some  of  the  higher  partials.  The  pianissimo  tones  of 
the  middle  register,  shown  on  the  second  Kne  of  the  figure, 
are  simple,  without  overtones. 

When  the  lower  register  is  played  forte,  it  is  in  effect  over- 
blown, and  the  first  overtone  becomes  the  most  prominent 
partial,  as  shown  on  the  third  line  ;  the  fundamental  is  weak, 


T 

3 


Fig.  140.    Analyses  of  flute  tones. 

being  just  loud  enough  to  characterize  the  pitch.  The 
player  is  often  conscious  of  the  skill  required  to  prevent  the 
total  disappearance  of  the  fundamental  and  the  passing  of 
the  tone  into  the  octave.  The  tones  of  the  low  register, 
when  played  loudly,  have  as  many  as  six  or  eight  partials, 
and  at  times  these  sounds  suggest  the  string  quality  of  tone, 
o  193 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


The  tones  of  the  middle  register  played  forte  consist  mainly 
of  fundamental  with  traces  of  the  second  and  third  partials. 
In  this  respect  these  flute  tones  are  very  similar  to  those  of 
the  soprano  voice  of  like  pitch,  a  fact  which  is  made  use 
of  in  the  duet  of  the  Mad  Scene  in  the  opera  ''Lucia  di 
Lammermoor." 

The  average  of  all  of  the  tones  of  the  lower  and  middle 
registers  of  the  flute,  shown  in  the  upper  line  of  the  figure, 
leads  to  the  conclusion  that  the  tone  of  the  flute  is  character- 
ized by  few  overtones,  with  the  octave  partial  predom- 
inating. 

The  tones  of  the  highest  register  have  been  analyzed  and 
found  to  be  practically  simple  tones.  This  result  is  to  be 
expected  from  the  conditions  of  tone  production  for  the 
higher  tones ;  the  air  column  is  of  a  diameter  relatively 
large  as  compared  with  its  length,  and  it  is  difficult  to  pro- 
duce loud  overtones  in  such  an  air  column. 

The  Violin 

The  strings  of  a  violin,  Fig.  141,  are  caused  to  \dbrate 
by  the  action  of  the  bow,  and  these  vibrations  are  trans- 
mitted through  the  bridge  and  body  of  the  instrument  to 
the  air ;  not  only  does  the  body  affect  the  air  by  its  surface 
movements,  but  the  interior  space  acts  as  a  resonance 
chamber. 

Helmholtz  has  made  a  study  of  the  vibrating  violin  string, 
and  has  developed  the  mathematical  equations  defining  the 
motion ;  Professor  H.  N.  Davis  has  investigated  the  lon- 
gitudinal vibrations  of  strings  in  a  manner  to  throw  much 
Ught  upon  the  subject ;  Professor  E.  H.  Barton  and  his 
colleagues  have  photographed  the  movements  of  the  string 
and  body  of  the  instrument ;      and  P.  H.  Edwards  and 

194 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


C.  W.  Hewlett  have  studied  the  tones  of  vioUns  of  differing 
quaUty.^- 

By  means  of  the  vibration  microscope  Hehnholtz  observed 
the  vibrations  of  the  string  and  plotted  the  form  of  its  move- 
ments, point  by  point,  as  shown  in  Fig.  142.  A  photograph 
of  a  sound  wave  from 
a  vioHn  is  given  in 
Fig.  143;  the  form 
is,  in  general,  iden- 
tical with  the  Helm- 
holt  z  diagram ;  this 
identitj^  is  remark- 
able when  it  is  re- 
membered that  the 
photograph  is  the 
wave  in  air,  from 
the  body  of  the  in- 
strument, while  the 
diagram  represents 
the  movements  of 
the  string.  The 
photograph  shows 
what  may  be  con- 
sidered the  typical 
form  of  a  \dolin 
wave,  but  it  is  not 
the  common  form  ; 

this  particular  shape  depends  upon  a  critical  relation  be- 
tween the  pressure,  grip,  and  speed  of  the  bow,  and  upon  the 
place  of  bowing  and  the  pitch  of  the  tone.  The  usual  varia- 
tions in  bowing  disturb  the  regularity  of  the  vibrations,  and 
produce  a  continually  changing  wave  form.    This  is  an 

195 


Fig.  141.    The  violin. 


THE  SCIENCE  OF  MUSICAL  SOUNDS 

indication  of  the  fact  that  a  great  variety  of  tone  quaUty 
can  be  produced  by  the  usual  changes  in  bowing.  BerHoz, 
in  describing  the  orchestral  usefulness  of  the  \dolin,  says : 
'^From  them  is  evolved  the  greatest  power  of  expression, 


Fig.  142.    Helmholtz's  diagram  of  the  vibrations  of  a  violin  string. 


and  an  incontestable  variety  of  qualities  of  tone.  Violins 
particularly  are  capable  of  a  host  of  apparently  inconsistent 
shades  of  expression.  They  possess  as  a  whole  force,  light- 
ness, grace,  accents  both  gloomy  and  gay,  thought,  and 


Fig.  143.    Photograph  of  the  tone  of  a  vioHn. 


passion.  The  only  point  is  to  know  how  to  make  them 
speak." 

The  tone  quality,  as  well  as  the  wave  form,  remains  con- 
stant so  long  as  the  bowing  is  constant  in  pressure,  speed, 
and  direction.    The  direction  of  bowing  may  be  skillfully 

196 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


reversed  without  changing  the  tone  quahty.  Fig.  144  is  a 
photograph  of  the  wave  form  when  a  change  of  bowing 
occurs ;  the  first  part  of  the  curve  is  for  an  up-bow,  while 
the  other  part  is  produced  by  the  down-bow ;  the  curve  is 
symmetrically  turned  over  wdth  every  change  in  the  direc- 
tion of  bowing,  while  the  confusion  caused  by  the  change 
produces  a  noise  which  lasts  about  two  hundredths  of  a 
second.  Analytically^,  the  turning  over  of  the  curve  means 
that  the  phases  of  all  the  components  are  reversed ;  the 
ear  does  not  detect  any  change  in  tone  quality  due  to  the 


Fig.  144.    Photograph  of  the  tone  of  a  violin  at  the  time  of  reversal  of  the  bowing. 

reversal  of  phases,  and  this  fact  supports  the  statement  that 
tone  quality  is  independent  of  phase. 

Photographs  were  taken  of  a  series  of  tones  on  each  string 
of  the  violin,  of  three  degrees  of  loudness.  The  average 
results  of  the  analysis  of  loud  tones  from  the  four  strings 
are  shown  in  Fig.  145.  For  the  lower  sounds  the  funda- 
mental is  weak,  as  indeed  it  must  be,  since  these  tones  are 
lower  than  the  fundamental  resonance  of  the  body  of  the 
violin ;  the  tones  from  the  three  higher  strings  have  strong 
fundamentals.  The  ear  perceives  a  fundamental  in  the 
lower  tones  of  the  violin,  and  this  must  result  from  a  beat- 
tone  produced  by  adjacent  higher  partials  which  are 
strong.  The  tones  from  the  three  lower  strings  seem 
to  be  characterized  by  strong  partials  as  high  as  the  fifth, 

197 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


while  the  E  string  gives  a  strong  third.  In  general  the 
tone  of  the  violin  is  characterized  by  the  prominence  of 
the  third,  fourth,  and  fifth  partials ;  and  while  the  violin 
generates  a  larger  series  of  partials  than  does  the  flute, 
yet  it  is  not  equal  to  the  brass  and  reed  instruments  in 
this  respect.     The  great  advantage  of  the  violin  over 


E  string 


1 


A  string   l_ 


J  L 


1 


J  1 


J  I  L 


I  \  \  \  \  1       I       I      I     I    I    I    I    I  I 

1  2  3         4       5      6    7    8   9  10  15 

Fig.  145.    Analyses  of  violin  tones. 

all  other  orchestral  instruments  in  expressiveness  is  due 
to  the  control  which  the  performer  has  over  the  tone 
production. 

The  Clarinet  and  the  Oboe 

The  study  of  reed  instruments  has  not  been  completed, 
but  the  analyses  of  many  individual  tones  show  interesting 
characteristics.  The  clarinet,  Fig.  146,  generates  sound 
by  means  of  a  single  reed  of  bamboo  which  \dbrates  against 

198 


TONE  QUALITIES  OF  MUSICAL  INSTRUiMENTS 


the  opening  in  the  mouthpiece ;  these  vibrations  are  con- 
trolled and  imparted  to  the  air  by  the  body  tube.  The 
body  has  a  uniform  cylindrical  bore,  at  the  lower  end  of 


Fig.  146.    The  clarinet.  Fig.  147.    The  oboe. 


which  is  a  short,  bell-shaped  enlargement.  The  keys  vary 
the  resonance  of  the  interior  column  of  air  and  thus  control 
the  pitch. 

The  oboe,  Fig.  147,  has  a  mouthpiece  consisting  of  two 
reeds  which  vibrate  against  each  other.    The  body  with  its 

]99 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


keys  forms  a  resonance  chamber  of  various  pitches,  but  it 
differs  from  the  clarinet  in  that  the  bore  is  conical  throughout. 


Fig.  148.    Photograph  of  the  tone  of  an  oboe. 


The  photograph  of  the  tone  from  an  oboe,  Fig.  148,  and 
that  from  a  clarinet,  Fig.  149,  both  show  deep  kinks  in  the 
wave  form ;  these  kinks  indicate  the  presence  of  relatively 


Fig.  149.    Photograph  of  the  tone  of  a  clarinet. 


very  loud  higher  partials  which,  no  doubt,  produce  the 
reedy  tone  quality  of  these  instruments.    The  presence 

200 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


of  beats,  that  is,  the  recurrence  in  each  wave  length  of  por- 
tions where  the  kinks  are  neutralized,  shows  that  there  are 
two  adjacent  high  partials  of  nearly  equal  strength,  as  was 
explained  under  Analysis  by  Inspection  in  Lecture  IV.  The 
average  of  several  analyses,  Fig.  150,  shows  that  the  oboe 
tone  has  twelve  or  more  partials,  the  fourth  and  fifth  pre- 
dominating, with  30  and  36  per  cent  respectively  of  the 
total  loudness.  The  clarinet  tone  may  have  twenty  or 
more  partials  ;  the  average  of  several  analyses  shows  twelve 
of  importance,  with  the  seventh,  eighth,  ninth,  and  tenth 
predominating ;  the  seventh  partial  contains  8  per  cent  of 

CLARINET  J  .  I  I  :  .  1  i  till.. 


J  I  t  t  t- 


I  2  3         4       5      6     7    8   9  10  15 

Fig.  150.    Analyses  of  the  tones  of  the  oboe  and  the  clarinet. 

the  total  loudness,  while  the  eighth,  ninth,  and  tenth  con- 
tain 18,  15,  and  18  per  cent  respectively. 

The  statement  is  often  made  that  the  seventh  and  ninth 
partials  are  ^ inharmonic"  and  that  their  presence  renders 
a  musical  sound  disagreeable.  The  seventh  and  ninth 
partials  are  just  as  natural  as  any  others;  a  partial  is  not 
inharmonic  because  it  is  the  seventh  or  ninth  in  the  series 
of  natural  tones ;  any  partial  whose  frequency  is  an  exact 
multiple  of  that  of  the  fundamental  is  truly  harmonic ; 
a  partial  is  inharmonic  when  it  is  not  an  exact  multiple 
of  the  fundamental  frequency,  whether  it  is  the  second  or 
ninth,  or  any  other  of  the  natural  series.  If  the  wave  form 
of  a  sound  is  periodic,  its  partials  must  all  be  harmonic, 

201 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


and  such  a  sound  is  musical.  The  clarinet  gives  periodic 
waves,  which,  as  the  analysis  shows,  contain  loud  seventh 
and  ninth  partials ;  these  partials  may  almost  be  said  to 
be  the  characteristic  of  the  tone,  but  they  are  in  tune,  and 
are  harmonic,  and  the  clarinet  tone  has  a  very  beautiful 
musical  quality.    Lanier  in  '^The  Symphony"  says 

"The  silence  breeds 
A  little  breeze  among  the  reeds 
That  seems  to  blow  by  sea-marsh  weeds ; 
Then  from  the  gentle  stir  and  fret 
Sings  out  the  melting  clarionet." 

The  adjective  melting"  seems  to  the  author  not  merely  a 
poetic  term,  but  a  real  description  of  the  clarinet  as  heard 
in  the  orchestra.. 

The  Horn 

The  horn,  Fig.  151,  is  a  brass  instrument  of  extreme  sim- 
plicity, consisting  of  a  slender  conical  tube,  sometimes 
more  than  eighteen  feet  long,  with  a  conical  cup-shaped 
mouthpiece,  and  a  large  flaring  bell.  In  its  typical  form 
there  are  no  apertures  in  the  walls  of  the  tube,  and  no  valves  ; 
but  the  modern  horn  usually  has  valves,  as  shown  in  the  figure. 

The  tone  of  the  horn  is  described  by  Lavignac  as  ''by 
turns  heroic  or  rustic,  savage  or  exquisitely  poetic ;  and  it 
is  perhaps  in  the  expression  of  tenderness  and  emotion  that 
it  best  develops  its  mysterious  qualities."  The  scien- 
tific analysis  shows  causes  for  the  variety  of  musical  effects, 
for  the  horn  produces  tones  of  widely  differing  composition 
from  one  as  soft  and  smooth  as  a  dehcate  flute  tone  to  a 
''split"  tone  that  is  tonally  disrupted  by  strong  higher 
partials.  The  low  sounds  of  the  horn  are  rich  in  overtones, 
containing  the  largest  number  of  partials  yet  found  in  any 
musical  tone.    The  analysis  of  the  wave  for  the  tone 

202 


TONE  QUALITIES  OF  MUSICAL  L\STRUMENTS 

D2  =  162,  Fig.  152,  shows  the  presence  of  the  entire  series  of 
partials  up  to  thirty,  with  those  from  the  second  to  the  six- 


FiG.  151.    The  horn. 


teenth  about  equally  loud ;  a  diagram  of  this  analysis  is 
given  on  the  lower  line  of  Fig.  153. 

The  results  of  analyses  of  various  other  tones  from  the 
horn  are  also  given  in  Fig.  153.    The  second  line  shows  the 


Fig.  152.    Photograph  of  the  tone  of  a  horn. 


average  composition  of  loud,  medium,  and  soft  tones  rang- 
ing over  the  entire  compass,  indicating  a  strong  fundamental 
followed  by  a  complete  series  of  partials,  more  than  twenty 

203 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


in  number,  of  gradually  diminishing  intensity.  The  tone 
quality  thus  indicated  approaches  more  nearly  to  the  ideal 
described  on  page  211  than  does  that  of  any  other  instru- 
ment so  far  investigated. 

A  ''chord  tone"  which  seems  to  be  made  by  humming 
with  the  vocal  chords  while  playing,  shown  in  the  third 
hne  of  the  figure,  has  the  fourth,  fifth,  and  sixth  partials  the 
most  prominent,  which  give  the  common  chord,  do,  mi,  sol 


SPUIT  _J  i  I  I  I  i  .  1     i  t 


LOW       _^  I  I  1  I  i  I  I  i  »    •    t    1   I  1  t 


I  \  \       \      \     \    !    I  I  I  I  I  I  I  I  I  I  lllllllllllll 

1  2  3         4       5      6     7    e   9  10  15        20      25  30 

Fig.  153.    Analyses  of  tones  of  the  horn. 

A  ''smooth"  tone  is  produced  when  the  player  muffles 
the  tone  more  or  less  by  putting  his  hand  in  the  bell  of  the 
horn.  The  analysis  of  such  a  tone,  shown  in  the  fourth 
line,  indicates  that  it  is  nearly  simple  and  is  very  much  like 
the  lower  tones  of  the  flute  when  played  softly. 

The  "rough"  tone  is  played  more  loudly  and  without 
muffling  by  the  hand ;  the  analysis,  line  five,  shows  an  oc- 
tave overtone  which  is  louder  than  the  fundamental,  and 
a  weak  third  partial. 

204 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


Another  quality  of  tone,  shown  on  the  top  Une,  is  called 
a  ''split  tone"  ;  this  tone  is  hterally  spht  into  man}'  partials 
and  distributed  uniformly  from  the  fundamental  to  the 
twelfth. 

The  Voice 

The  sounds  of  the  voice  originate  in  the  vibrations  of  the 
vocal  cords  in  the  larynx,  the  pitch  being  controlled  largely 
by  muscular  tension,  while  the  quahty  is  dependent  mostly 
upon  the  resonance  effects  of  the  vocal  cavities. 

The  tones  of  the  singing  voice  have  not  been  analyzed 
except  in  connection  with  the  vowels,  the  results  of  which 
are  described  at  length  in  Lectures  VII  and  VIII.  Fig. 
154  shows  the  curve  for  a  bass  voice  (E  C)  intoning  the 


vowel  a  in  father  on  the  note  Fi =  92, 


The  loud 


partials  in  this  tone  are  evidently  of  a  high  order,  since 


Fig.  154.    Photograph  of  a  bass  voice. 

there  are  many  large  kinks  in  one  wave  length.  A  diagram 
of  the  analysis  of  the  curve  is  given  in  the  lower  line  of  Fig. 
156 ;  it  shows  that  the  seventh,  eighth,  ninth,  and  eleventh 
partials  are  the  strongest. 

The  voice  E  E  AI  intoned  the  same  vowel  on  the  soprano 

pitch  of  Bp, 


producing  the  curve  shown  in  Fig. 

155,  the  analysis  of  which  is  given  in  the  upper  line  of 

205 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Fig.  156.    This  curve  is  simple,  and  is  seen  at  a  glance  to  con- 
tain but  one  strong  partial,  the  second  over  tone  or  octave. 
These  two  voices,  and  their  corresponding  curves,  are 


Fig.  155.    Photograph  of  a  soprano  voice. 


very  unhke,  yet  the  ear  recognizes  the  same  vowel  from 
both.  The  vowel  characteristics  of  the  bass  voice,  repre- 
sented by  the  seventh,  eighth,  ninth,  and  eleventh  partials, 
are  accompanied  by  six  lower  partials,  the  first  of  which 


; 

; 

2 

\ 

1 

E 

c 

1 

\ 

i 

1 

C,   D    EF    C    A    BC,  D    EF    OA    BC3  D    E  F    G    A    B       D    E  F    G    A    BC5   D    E  F    G    A    B  Cj   D    E  F    G   A    BC7  D  E 

65  otP.^MeNT  or  P»,s,c.  129,.s.s=Ho,  =>c=  259  517  1035  2069  4138 

Fig.  156.    Analyses  of  tones  of  bass  and  soprano  voices. 

determines  the  pitch  while  the  others  give  the  bass  quality 
of  the  individual  voice.  The  vowel  characteristic  for  the 
other  voice  is  the  second  partial,  the  pitch  is  determined 
by  the  first  or  fundamental,  while  a  series  of  five  or  more 
higher  partials  produce  the  individuality  of  the  soprano  tone. 

206 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


The  Piano 

The  vibrations  in  a  piano  originate  in  the  strings  which 
are  struck  with  the  felt-covered  hammers  of  the  key  action, 
while  the  sound  comes  mostly  from  the  soundboard.  The 
relations  of  strings  and  soundboard  have  been  considered 
under  Resonators  and  Resonance  in  the  beginning  of  this 
lecture. 

The  lower  tones  of  the  piano  are  found  to  be  very  weak 
in  fundamental,  but  to  have  many  overtones,  partials  as 
high  as  the  forty-second  having  been  identified.  These 
high  partials  are  loud  enough  to  be  heard  by  the  unaided 
ear  after  attention  has  been  directed  to  them.  These 
characteristics  are  entirely  consistent  with  the  nature  of  the 
source,  which  is  a  slender  metal  string  struck  with  a  hammer. 

The  higher  tones  of  the  piano,  originating  in  much  shorter 
strings  under  high  tension,  have  few  partials,  and  the  loud- 
est component  is  often  the  second  partial  or  octave.  The 
tones  from  the  middle  portion  of  the  scale  contain  ten  or 
more  partials  of  well  distributed  intensity. 

The  piano  is  perhaps  the  most  expressive  instrument, 
and  therefore  the  most  musical,  upon  which  one  person 
can  play,  and  hence  it  is  rightly  the  most  popular  instru- 
ment. The  piano  can  produce  wonderful]  varieties  of 
tone  color  in  chords  and  groups  of  notes,  and  its  music 
is  full,  rich,  and  varied.  The  sounds  from  any  one  key 
are  also  susceptible  of  much  variation  through  the  nature 
of  the  stroke  on  the  key.  So  skillful  does  the  accom- 
plished performer  become  in  producing  variety  of  tone 
quality  in  piano  music,  which  expresses  his  musical  moods, 
that  it  is  often  said  that  something  of  the  personality  of 
the  player  is  transmitted  by  the  ''touch"  to  the  tone 

207 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


produced,  something  which  is  quite  independent  of  the 
loudness  of  the  tone.  It  is  also  claimed  that  a  variety  of 
tone  qualities  may  be  obtained  from  one  key,  by  a  vari- 
ation in  the  artistic  or,  emotional  touch  of  the  finger 
upon  the  key,  even  when  the  different  touches  all  pro- 
duce sounds  of  the  same  loudness.  This  opinion  is  al- 
most universal  among  artistic  musicians,  and  doubtless 
honestly  so.  These  musicians  do  in  truth  produce  marvel- 
ous tone  qualities  under  the  direction  of  their  artistic 
emotions,  but  they  are  primarily  conscious  of  their  personal 
feelings  and  efforts,  and  seldom  thoroughly  analyze  the 
principles  of  physics  involved  in  the  complicated  mechanical 
operations  of  tone  production  in  the  piano.  Having  investi- 
gated this  question  with  ample  facilities,  we  are  compelled 
by  the  definite  results  to  say  that,  if  tones  of  the  same  loud- 
ness are  produced  by  striking  a  single  key  of  a  piano  with 
a  variety  of  touches,  the  tones  are  always  and  necessarily 
of  identical  quality ;  or,  in  other  words,  a  variation  of  artis- 
tic touch  cannot  produce  a  variation  in  tone  quality  from 
one  key,  if  the  resulting  tones  are  all  of  the  same  loudness. 
From  this  principle  it  follows  that  any  tone  quality  which 
can  be  produced  by  hand  playing  can  be  identically  repro- 
duced by  machine  playing,  it  being  necessary  only  that 
the  various  keys  be  struck  automatically  so  as  to  produce 
the  same  loudness  as  was  obtained  by  the  hand,  and  be 
struck  in  the  same  time  relation  to  one  another.  There 
are  factors  involved  in  the  time  relations  of  beginning  the 
several  tones  of  a  chord  or  combination,  which  are  not  often 
taken  into  account ;  a  brief  notice  of  the  nature  of  piano 
tone  will  enable  us  to  establish  this  conclusion. 

Two  photographs  of  piano  tones  are  shown,  the  first, 
Fig.  157,  being  of  the  note  one  octave  above  middle  C 

208 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 

and  the  other,  Fig.  158,  of  the  note  one  octave  below. 
The  first  photograph  shows  two  innportant  features :  the 
sound  rises  to  its  maximum  intensity  in  about  three  one- 


FiG.  157.    Photograph  of  the  tone  of  a  piano. 


hundredths  of  a  second,  and  in  one  fifth  of  a  second  it 
has  fallen  to  less  than  a  tenth  of  its  greatest  loudness ; 
it  then  gradually  dies  out,  but  with  a  progressive  change 
in  quality.     In  the  beginning  the  fundamental  is  the 


Fig.  158.    Photograph  of  the  tone  of  a  piano. 


loudest  component,  but  after  a  tenth  of  a  second,  the  octave 
is  the  loudest  part. 

The  second  photograph  is  of  a  tone  two  octaves  lower  and 
is  of  a  much  more  complex  nature.    There  are  more  than 
p  209 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


ten  partials  of  appreciable  loudness,  which  are  continually 
changing  in  relative  intensity,  due,  no  doubt,  to  peculiarities 
of  piano  construction  which  prolong  certain  partials  and 
absorb  others.  Whatever  complex  tone  may  be  generated 
by  the  hammer  blow,  the  quality  of  tone  that  enters  into 
combination  with  that  from  other  strings  is  dependent 
upon  the  parts  of  the  tones  from  the  several  strings  being 
simultaneously  coexistent.  The  quality  of  tone  obtained 
from  a  piano  when  a  melody  note  is  struck  is  dependent 
upon  the  mass  of  other  tones  then  existing  from  other  keys 
previously  struck  and  sustained,  and  it  depends  upon  the 
length  of  time  each  of  these  tones  has  been  sounding.  It 
is  evident  that  not  only  does  a  piano  give  great  variety  of 
tone  by  various  degrees  of  hammer  blow,  but  there  is  pos- 
sible an  almost  infinite  variety  of  tone  quality  in  combina- 
tions of  notes  struck  at  intervals  of  a  few  hundredths  of  a 
second.  It  is  beheved  that  the  artistic  touch  consists  in 
slight  variations  in  the  time  of  striking  the  different  keys, 
as  well  as  in  the  strength  of  the  blow,  and  that  tone  quality 
is  determined  by  purely  physical  and  mechanical  considera- 
tions. 

The  correctness  of  this  argument  is  further  supported  by 
the  mechanical  piano  players,  which  attempt  to  reproduce 
the  characteristics  of  individual  pianists.  The  more  highly 
developed  such  instruments  become,  the  more  nearly  they 
imitate  hand  playing  in  musical  effects ;  in  many  instances 
the  imitation  is  practically  perfect,  and  I  believe  that  in 
the  near  future  the  automatic  piano  will  reproduce  all  of 
the  effects  of  hand  playing. 

This  condition  will  in  no  way  displace  the  artist,  nor  will 
it  in  the  least  reduce  his  prestige ;  on  the  contrary,  it  will 
enhance  his  standing,  and  we  shall  honor  him  the  more  for 

210 


TONE  QUALITIES  OF  MUSICAL  INSTRUMENTS 


his  accomplishments.  The  machine  can  never  create  a 
musical  interpretation,  the  artist  must  ever  do  this. 

Sextette  and  Orchestra 

An  illustration  of  very  complex  tone  quality  is  obtained 
with  the  talking  machine  reproducing  the  Sextette  from 
"  Lucia  di  Lammermoor/'  by  six  famous  voices  with  orches- 
tral accompaniment ;  photogi'aphs  of  small  portions  of  this 
music  are  shown  in  the  frontispiece.  The  dots  on  the  lower 
edge  of  the  picture  are  time  signals  which  are  second 
apart ;  each  line  of  the  picture  represents  the  \dbrations 
due  to  music  of  less  than  one  second's  duration.  On  the 
scale  of  the  original  photograph,  which  is  five  inches  wide, 
the  length  of  film  required  to  record  the  entire  selection  would 
be  1000  feet.  The  effects  impressed  upon  the  wave  by  a 
particular  voice  or  instrument  are  clearly  reproduced :  in 
the  middle  of  the  top  hne,  the  increase  in  the  amphtude 
of  the  wave  is  due  to  the  entrance  of  the  tenor  voice ;  the 
second  line  shows  the  comparatively  simple  wave  of  the  solo 
soprano  voice  singing  high  Bb,  the  smoothness  of  the  curve 
attesting  the  pure  quality  of  the  voice. 

The  Ideal  ^Musical  Toxe 

Neither  science  nor  art  furnishes  criteria  which  will 
define  the  ideal  musical  tone ;  a  scientific  investigation 
and  analysis  of  the  sound  from  a  violin  or  a  piano  cannot 
determine  whether  it  is  the  ideal.  ]\lusical  instruments  are 
used  for  artistic  purposes  and  their  selection  is  ultimately 
determined  by  the  aesthetic  taste  of  the  artist.  WTien  an 
instrument  has  been  artistically  approved,  the  physicist 
can  describe  its  tonal  characteristics  and  select  other  instru- 
ments possessing  the  same  qualities ;  he  can  detect  defi- 

211 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


ciencies  and  defects  and,  perhaps,  can  suggest  remedies. 
''The  chemist  can  scrape  the  paint  from  a  canvas  and  ana- 
lyze it,  but  he  cannot  thereby  select  a  masterpiece." 

A  musical  tone  of  remarkable  quality  may  be  produced 
by  a  special  set  of  ten  tuning  forks  shown  in  Fig.  159 ;  these 
forks  are  accurately  tuned  to  the  pitches  of  a  fundamental 
tone  of  128  vibrations  per  second  and  its  nine  harmonic 
overtones.     When  the  fundamental  alone  is  sounding,  a 


Fig.  159.    Set  of  tuning  forks  for  demonstrating  the  quality  of  composite  tones. 

sweet  but  dull  tone  is  heard.  As  the  successive  overtones 
are  added,  the  tone  grows  in  richness,  until  the  ten  forks 
are  sounding,  when  the  effect  is  that  of  one  splendid 
musical  tone.  One  is  hardly  conscious  that  the  sound 
is  from  ten  separate  sources,  the  components  blend  so 
perfectly  into  one  sound.  The  tone  is  vigorous  and  ^4iving" 
and  has  a  fullness  and  richness  rarely  heard  in  musical 
instruments. 

Bearing  in  mind  the  qualifications  just  mentioned, 
one  may  speak  of  an  ideal  musical  tone,  meaning  the 
most   gratifying  single  tone    which    can    be  produced 

212 


TONE  QUALITIES  OE  Ml  SICAL  INSTRUMENTS 


from  one  instrument.  Following  the  above  experiment 
the  ideal  tone  may  be  arbitrarily  described  as  one 
having  a  strong  fundamental  containing  perhaps  50  per 
cent  of  the  total  intensity,  accompanied  by  a  complete 
series  of  twenty  or  more  overtones  of  successively  dimin- 
ishing intensity. 

If,  while  the  forks  in  the  above  experiment  are  sounding, 
they  are  silenced  in  succession  from  the  highest  downward, 
the  tone  becomes  less  and  less  rich,  until  finally  the  funda- 
mental alone  is  heard.  This  is  a  simple  tone  and  is  of  a 
dull,  droning  quality ;  the  experiment  demonstrates  that 
a  pure  tone  is  a  poor  tone. 

It  is  by  no  means  desirable  that  all  musical  instruments 
should  have  the  quality  of  tone  described.  The  great 
variety  of  musical  tone  coloring  obtained  by  the  modern 
composers  requires  instruments  of  the  greatest  possible 
divergence  in  quality  ;  the  contrasts  thus  available  are  very 
effective.  Of  the  instruments  of  the  orchestra,  perhaps  the 
horn,  in  certain  of  its  lower  tones,  approaches  most  nearly 
to  the  arbitrary  ideal. 

Demonstration 

In  the  oral  lectures,  the  characteristics  of  various  instru- 
ments as  described  in  the  preceding  pages  and  as  shown  by 
the  photographs,  were  demonstrated  by  playing  the  instru- 
ments themselves  before  the  phonodeik,  which  projected 
the  sound  waves  upon  the  screen  as  explained  in  Lecture 
III.  The  sounds  so  demonstrated  were  the  simple  and 
complex  tones  from  tuning  forks,  the  flute  tone  as  it  develops 
from  the  simple  pianissimo  quality  to  the  more  complex 
fortissimo  by  the  addition  of  successive  overtones,  the  full 
and  vibrant  cornet  tone  having  many  partials,  the  string 

213 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


tone  of  the  violin,  with  the  reversal  of  phases  by  changing 
the  direction  of  bowing,  the  reedy  tone  of  the  clarinet,  the 
varying  qualities  of  vocal  tones,  the  clanging  tone  from  a 
bell  with  its  interfering  inharmonic  overtones,  and  finally 
the  vocal  sextette  with  orchestra,  and  the  concert  band,  as 
reproduced  by  various  types  of  phonographs. 

As  seen  upon  the  screen,  the  waves  of  light,  which  may 
be  ten  feet  vnde  and  forty  feet  long,  stretching  across  the 
end  of  the  room,  are  constantly  in  motion,  and  pass  from 
one  wave  form  to  another,  from  simple  to  most  complex 
shapes,  with  every  change  in  frequency,  loudness,  or  quality 
of  the  sound ;  the  wonderfully  changing  waves  flow  with 
perfect  smoothness  and  reproduce  visually  the  harmoni- 
ously blending  movements  of  the  air,  which  the  ear  inter- 
prets as  music.  This  ability  to  see  the  effects  of  quahtative 
changes  as  well  as  to  hear  them  is  certainly  advantageous 
in  an  analytical  study  of  sounds,  and  possibly  it  adds  to  the 
musical  effectiveness ;  it  is  at  least  a  fascinating  and  instruc- 
tive demonstration. 


214 


LECTURE  VIl 


PHYSICAL  CHARACTERISTICS  OF  THE  VOWELS 
The  Vowels 

The  vowels  have  been  more  extensively  investigated 
than  any  other  subject  connected  ^vith  speech ;  the  philolo- 
gist, the  physiologist,  the  physicist,  and  the  vocalist,  has 
each  attacked  the  problem  of  vowel  characteristics  from 
his  own  separate  point  of  view.  The  methods  of  the  sev- 
eral classes  of  investigators,  and  the  expressions  of  the  re- 
sults, are  so  unlike  and  so  highly  specialized,  that  one 
person  is  seldom  able  to  appreciate  them  all. 

The  physicist  wishes  to  interpret  the  vowels  as  they  exist 
in  the  sound  waves  in  air,  that  is,  he  wishes  to  know  the 
nature  of  the  musical  tone  quality  which  gives  individuality 
to  the  several  vowels.  The  tone  quality  of  vowels  has  been 
more  closely  studied  than  that  of  all  other  sounds  combined, 
and  yet  no  single  opinion  of  the  cause  of  vowel  quality  has 
prevailed. 

The  first  attempt  at  an  explanation  of  vowel  quahty  was 
made  in  1829  by  Willis,  who  concluded  from  experiments 
with  reed  organ  pipes  that  it  depends  upon  a  fixed  charac- 
teristic pitch ;  this  theory  was  extended  by  Wheatstone 
(1837)  and  by  Grassmann  (1854).  Bonders  (1864)  dis- 
covered that  the  cavity  of  the  mouth  is  tuned  to  different 
pitches  for  different  vowels.  Helmholtz  (1862-1877)  ex- 
pounded the  theory,  a  development  of  those  given  before, 

215 


THE  SC  IENCE  OF  MUSICAL  SOUNDS 


that  each  vowel  is  characterized,  not  by  a  single  fixed  pitch, 
but  by  a  fixed  region  of  resonance,  which  is  independent 
of  the  fundamental  tone  of  the  vowel ;  this  is  the  so-called 
fixed-pitch  theory. 

In  opposition  to  this  theory,  many  writers  on  the  sub- 
ject have  held  that  the  quality  of  a  vowel,  as  well  as  that 
of  a  musical  instrument,  is  characterized  by  a  particular 
series  of  overtones  accompanying  a  given  fundamental, 
the  pitches  of  the  overtones  varying  with  that  of  the  funda- 
mental, so  that  the  ratios  remain  constant ;  this  is  the 
relative-pitch  theory. 

Auerhach  in  1876  developed  an  intermediate  theory, 
concluding  that  both  characteristics  are  concerned,  and 
that  the  pitch  of  the  most  strongly  reinforced  partial  alone 
is  not  sufficient  to  determine  the  vowel.  Hermann  (1889) 
has  suggested  that  the  vowels  might  be  characterized  by 
partial  tones,  the  pitches  of  which  are  within  certain  limits, 
but  which  are  inharmonic,  the  partials  being  independent  of 
the  fundamental.  Lloyd  (1890)  considers  that  the  identity 
of  a  vowel  depends  not  upon  the  absolute  pitch  of  one  or 
more  resonances,  but  upon  the  relative  pitches  of  two  or  more. 

Several  quotations  will  indicate  the  uncertainty  existing 
at  the  present  time  in  regard  to  the  nature  of  the  vowels. 
Ellis,  the  translator  of  Helmholtz,  writes *(1885)  :  ''The  ex- 
treme divergence  of  results  obtained  by  investigators  shows 
the  inherent  difficulties  of  the  determination."  Lord 
Rayleigh  (1896)  says  :  ''A  general  comparison  of  his  results 
with  those  obtained  by  other  methods  has  been  given  by 
Hermann,  from  which  it  will  be  seen  that  much  remains 
to  be  done  before  the  perplexities  involving  the  subject 
can  be  removed."  Auerhach  (1909)  discusses  the  various 
theories,  but  without  deciding  which  is  correct. 

210 


PHYSICAL  CHARACTERISTICS  OF  THE  VOWELS 


Two  recent  publications  on  this  subject  arrive  at  opposite 
conclusions.  Professor  Bevier  of  Rutgers  College  in  one  of 
the  most  complete  studies  yet  made  (1900-1905),^^  using 
the  phonograph  as  an  instrument  of  analysis,  arrives  at 
conclusions  in  accord  with  Helmholtz's  fixed-resonance 
theory  and  the  method  of  harmonic  analysis.  Professor 
Scripture  (1906),  formerly  of  Yale  University,  says:  ''the 
overtone  theor>^  of  the  vowels  cannot  be  correct"  ;  and  he 
gives  extended  arguments  in  support  of  this  opinion  and 
opposed  to  harmonic  analysis  of  vowels.^'  The  results  of 
the  work  here  described  are  in  entire  agreement  \\ith  Helm- 
holtz's  theory,  and  they  are,  therefore,  out  of  harmony  with 
Scripture's  arguments. 

Standard  Vowel  Tones  and  Words 

Vowels  are  speech  sounds  which  can  'be  continuously 
intoned,  separated  from  the  combinations  and  noises  by 
which  they  are  made  into  words.  A  dictionary  definition 
of  a  vowel  is:  "one  of  the  openest,  most  resonant,  and 
continuable  sounds  uttered  by  the  voice  in  the  process  of 
speaking ;  a  sound  in  which  the  element  of  tone  is  pre- 
dominant ;  a  tone-sound,  as  distinguished  from  a  fricative 
(rustling  sound),  from  a  mute  (explosive),  and  so  on." 

Helmholtz  specifies  seven  vowels,  the  'Century  Diction- 
ary "  gives  nineteen  vowel  sounds  in  its  key  to  pronunciation, 
w^hile  some  ^Titers  on  phonetics  tabulate  as  many  as  seventy- 
two  vowel  sounds.  After  preliminary  study,  eight  standard 
vowels  contained  in  the  follomng  words  were  selected  for 
definitive  analysis  :  father,  raw,  no,  gloom,  mat,  pet,  they, 
and  hee. 

The  particular  vowels  specified  are  according  to  the  pro- 
nunciation of  the  author.    It  must  be  remembered  that 

217 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


any  change  in  the  pronunciation  produces  a  different  vowel, 
though  we  may  understand  the  word  to  be  the  same,  and 
that  the  quantitative  results  would  vary  for  the  slightest 
change  in  intonation  or  inflection.  Since  individual  pro- 
nunciations vary  greatly,  even  within  the  range  of  one 
language,  there  seems  to  be  no  better  method  of  defining 
a  vowel  than  by  specifying  several  words,  in  each  of  which 
the  author  gives  the  vowel  the  same  sound.  Others  may 
disagree  with  some  of  the  pronunciations,  but  this  does 
not  change  the  fact  that  these  are  the  sounds  studied  and 
defined  in  the  results.  A  table  of  such  words  follows,  while 
a  larger  list  is  given  on  page  257. 

father,  far,  guard 
raw,  fall,  haul 
^no,  rode,  goal 
gloom,  move,  group 
mat,  add,  cat 
pet,  feather,  bless 
they,  bait,  hate 
bee,  pique,  machine 

Some  of  these  sounds  are  common  to  all  languages ;  the 
equivalent  of  father  is  found  in  German  in  vater,  and  in 
French  in  pate;  the  equivalent  of  no  in  German  is  in  wohl, 
and  in  French  in  cote;  but  there  seems  to  be  no  equivalent 
in  either  German  or  French  for  raw  or  mat. 

For  the  sake  of  simplicity,  instead  of  using  single  letters 
in  connection  with  a  multiplicity  of  signs  to  designate  the 
several  vowels,  the  writer  will  give  the  whole  word  con- 
taining the  vowel,  the  latter  being  indicated  by  italics  ; 
in  pronouncing  the  phrase  "si  record  of  the  vowel  father," 
one  may  emphasize  and  prolong  the  vowel  as  '^a  record  of 

218 


PHYSICAL  CHARACTERISTICS  OF  THE  VOWELS 


the  vowel  iah  .  .  .  ther,"  or,  better,  one  may  pronounce  only 
the  vowel  part  of  the  last  word,  as  ''a  record  of  the  vowel 
.  .  .ah  " 

Photographing,  Analyzing,  and  Plotting  Vowel 

Curves 

The  general  procedure  in  the  investigation  of  a  vowel  is 
as  follows  :  the  speaker  begins  to  pronounce  the  appropriate 
word  and  prolongs  the  vowel  in  as  natural  a  manner  as 


possible  ;  b}'  means  of  the  phonodeik  a  photographic  record 
is  taken  of  the  central  portion  of  the  vowel,  while  the  zero 
line  and  time  signals  are  recorded  simultaneously  with  the 
voice  curve.  The  vowel  curve  is  then  analyzed  into  its 
harmonic  components,  corrections  are  applied,  percentage 
intensities  for  the  several  partials  are  computed,  and  the 
results  are  diagramed,  as  explained  in  Lecture  V. 

A  photograph  of  the  vowel  father  intoned  by  a  baritone 
voice,  at  the  pitch  of  F2  =  182,  is  shown  in  Fig.  160.  The 
analysis  of  this  curve  is  given  in  Fig.  129,  page  169,  while 
analyses  of  other  photographs  of  the  same  vowel  are  shown 
in  Figs.  161,  162,  and  163. 


Fig.  160.    Photograph  of  the  vowel  a  in  father,  for  analysis. 


219 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


The  ordinates  on  a  vowel  diagram  indicate  the  distri- 
bution of  the  energy  of  the  sound  with  reference  to  its 
own  harmonic  partials.  A  single  analysis  gives  little 
information  as  to  the  distribution  of  energy  for  sounds  of 
intermediate  pitches,  since  the  sound  analyzed  can  have 
no  intensity  whatever  for  pitches  other  than  those  of  its 
own  partials.  If  the  vowel  is  intoned  by  the  same  person 
at  a  different  pitch,  its  partials  may  lie  between  those  of  the 


I 

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10 

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15 

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6 

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7 

B 

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EF 

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G 

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BC 

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EF 

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EF 

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»^2"c-t    I    I    11'  III    11    ii^.T.Ti  .T.T.TT.  T.Ti  .1   T   iri    TT    1   T   iri  T 

(|.|!Ni|;Nils|^l4ia^|i|il«l«fefelVlifelsl^ 

129  259  517  1035  2069  4138 


Fig.  161.    Loudness  of  the  several  components  of  the  vowel  father,  intoned  at 

two  different  pitches. 

first  sound  as  shown  in  Fig.  161.  By  plotting  many 
analyses  to  one  base  hne,  Fig.  162,  D,  a  curve  can  be  drawn 
which  shows  the  resonance  of  the  vocal  cavities  for  the 
particular  vowel.  For  purposes  of  analytical  study  it 
is  permissible  to  show  the  relations  of  the  separate  points 
of  a  single  analysis  to  the  indicated  resonance  curve  as  is 
done  in  A,  B,  and  C,  Fig.  162.    The  significance  of  these 

220 


PHYSICAL  CHARACTERISTICS  OF  THE  VOWELS 


curves  is  more  fully  explained  in  the  section  on  Classifica- 
tion of  the  Vowels,  on  page  228. 

Vowels  of  Various  Voices  and  Pitches 

Each  of  the  eight  vowels  has  been  photographed  at  several 
pitches  as  intoned  by  each  of  eight  voices,  giving  about  a 
thousand  curves,  all  of  which  have  been  analyzed  and 


129  259  517  1035  2069  4138 

Fig.  162.    Distribution  of  energy  among  the  several  partials  of  the  vowel  father, 
intoned  at  various  pitches. 

plotted.  There  were  two  bass  voices,  two  baritones,  one 
tenor,  one  contralto,  one  boy  soprano,  and  one  girl  soprano ; 
the  normal  pitches  of  these  voices  ranged  from  106  to  28L 
The  vowel  father  was  intoned  by  the  voice  D  C  M  at 
the  pitch  D2j^  =  155,  its  energy  distribution  curve  being 
as  shown  in  the  lower  part  of  Fig.  162  ;  the  next  two  curves, 
B  and  C,  show  the  same  vowel  by  the  same  voice  intoned 
at  pitches  of  F2S  =  182,  and  AsJ}  =  227.    When  the  vowel 

221 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


is  intoned  at  the  lowest  pitch,  the  sixth  partial  having  a  fre- 
quency of  930  contains  69  per  cent  of  the  total  energ}^  of 
the  sound ;  in  the  second  case  the  fifth  partial  of  pitch  910 
is  loudest  with  48  per  cent  of  the  energy ;  while  in  the  third 
case  the  fourth  partial  of  pitch  908  contains  65  per  cent  of 
the  energy. 


1 

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A   BC2  D   EF    G   A   BC3'  D   EF    G   A    BC4   D   EFGABC5D    E  F    G    A    B  Cg  D   E  F    G   A  BC7 
129  259  517  1035  2069  4138 


Fig.  163.    Distribution  of  energy  among  the  several  partials  of  the  vowel  father, 
as  intoned  by  eight  different  voices. 

The  same  vowel,  father,  was  intoned  by  the  same  voice, 
D  C  M,  approximately  upon  each  semitone  of  the  octave 
from  C2  =  129  to  C3  =  259,  at  twelve  different  pitches ; 
the  upper  part  of  the  figure,  D,  shows  the  location  of  all 
the  component  intensities  of  the  twelve  analyses ;  instead 
of  twelve  separate  curves,  one  is  drawn  showing  the  average 
energy  distribution. 

The  energy  curves  of  the  same  vowel,  father,  intoned 

222 


PHYSICAL  CHARACTERISTICS  OF  THE  VOWELS 


by  eight  different  voices,  at  pitches  ranging  from  106  to 
522,  are  given  in  Fig.  163.  The  voices  are  a  bass  (0  F  E), 
a  bass  (E  C),  a  baritone  (W  R  W),  a  baritone  (D  C  M),  a 
tenor  (F  P  W),  a  contralto  (E  E  M),  a  boy  soprano  (K  S, 
14  years  old),  and  a  girl  soprano  (H  F,  10  years  old). 

The  energ}^  curves  for  the  vowel  hee,  intoned  by  the  same 
eight  voices,  at  pitches  ranging  from  111  to  -400,  are  shown 


[29  259  517  1035  2069  41S8 

Fig.  164.    Distribution  of  energy  among  the  several  partials  of  the  vowel  hec, 
as  intoned  by  eight  different  voices. 


in  Fig.  164.  For  this  vowel  there  are  two  regions  of  reso- 
nance, one  at  a  pitch  of  about  300,  and  the  other  at  a  pitch 
of  about  3000.  \Miile  the  greater  part  of  the  energy-  is  in 
the  lower  resonance,  yet  it  may  be  said  that  the  higher 
resonance  is  the  characteristic  one,  since  its  absence  con- 
verts the  vowel  hee  into  gloom,  as  described  on  page  231. 

These  diagrams  indicate  that  there  is  not  a  fixed  partial 
wliich  characterizes  the  vowel,  neither  is  there  a  single, 

223 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


fixed  pitch.  The  greater  part  of  the  energy  of  the  voice  is 
in  those  partials  which  fall  within  certain  limits,  no  matter 
at  what  pitch  the  vowel  is  uttered,  nor  by  what  quality 
of  voice;  that  is,  the  vowel  is  characterized  by  a  fixed 
region,  or  regions,  of  resonance  or  reinforcement.  To  estab- 
lish this  theory  it  is  necessary  to  show  that  all  the  different 
vowels  have  distinctly  different  characteristic  regions  of 
resonance,  which  remain  the  same  for  all  voices.  An  in- 
vestigation has  been  made  leading  to  this  conclusion,  a 
detailed  account  of  which  is  to  be  published  elsewhere ; 
the  nature  of  this  study  is  indicated  by  the  following  de- 
scription, which  refers  to  one  voice  only. 

Definitive  Investigation  of  one  Voice 

The  study  of  the  vowels  of  different  voices  and  pitches 
showed  that  it  is  practically  impossible  to  obtain  the  same 
vowel  from  the  various  voices  with  sufficient  certainty  to 
permit  of  a  definitive  study,  and  even  extreme  variations  in 
pitch  for  one  voice  probably  alter  the  accuracy  of  pronuncia- 
tion. Therefore  it  was  decided  to  make  a  final  study  of 
the  principal  vowel  tones  of  the  English  language  as  spoken 
by  one  person  in  order  to  determine  the  physical  cause  of 
their  differences. 

Each  of  the  eight  vowels  previously  mentioned  was 
intoned  by  one  voice,  D  C  M ;  six  photographs  were  made 
in  succession  of  the  vowel  spoken  in  normal  pitch  and  inflec- 
tion ;  then  six  more  photographs  for  the  same  vowel  were 
made,  but  approximately  on  six  equidistant  tones  covering 
one  octave,  beginning  a  little  below  normal  pitch.  Thus 
twelve  curves  were  obtained  for  each  vowel,  and  for  some 
a  larger  number  was  made.  There  are  202  photographs  in 
this  second  series,  all  being  made  under  exactly  the  same 

224 


PHYSICAL  CHARACTERISTICS  OF  THE  VOWELS 

conditions  of  the  speaker's  voice  and  recording  apparatus. 
The  curves  were  analyzed  and  reduced  in  one  group,  a  sepa- 
rate energy  distribution  curve  was  drawn  for  each  analysis, 
and  finally  a  composite  or  average  curve  was  made  for  each 
vowel. 

Classification  of  Vowels 

The  eight  final  composite  vowel  curves,  drawn  on  separate 
pieces  of  paper,  were  arranged  upon  a  table  and  their  pe- 
culiarities studied.  Many  schemes  of  classification  were 
tried,  with  the  final  conclusion  that  all  vowels  may  be  divided 
into  two  classes,  the  first  having  a  single,  simple  charac- 
teristic region  of  resonance,  while  for  the  second  there  are 
two  characteristic  regions. 

The  vowels  of  the  first  class  are  represented  by  father, 
raw,  no,  and  gloom.  The  investigations  indicate  that  the 
most  natural  vowel  sound  and  the  most  elemental  words 
used  in  speech  are  ma  and  pa,  and  one  of  these  may  be 
selected  as  a  starting  point  for  a  classification.  It  adds  to 
the  effectiveness  if  the  vowels  are  indicated  by  simple 
syllables  of  the  same  general  form ;  for  the  vowels  of  the 
first  class  the  words  may  be  ma,  mai^,  mow,  and  moo,  or  pa, 
paw,  poe,  and  pooh. 

The  vowels  of  the  second  class  are  represented  by  mat, 
pet,  they,  and  hee ;  and  the  new  syllables  selected  are  mat, 
met,  mate,  and  meet,  or  pat,  pet,  pate,  and  peat.  This 
series  may  be  presumed  to  start  from  the  fundamental 
vowel  ma,  which  for  similarity  may  be  expressed  by  the 
words  mot  or  pot. 

The  characteristic  curves  for  vowels  of  the  first  class 
are  shown  in  Fig.  165 ;  the  vowels  are  ma,  maw,  mow,  and 
moo,  having  maximum  resonances  at  pitches  of  910,  732, 
461,  and  362,  respectively.  The  resonance  regions  overlap 
Q  225 


THE  SCIENCE  OF  MUSICAL  SOUNDS 

but  little ;  the  partials  lying  within  a  characteristic  region 
of  resonance  often  contain  as  much  as  90  per  cent  of  the 
total  energy  of  the  sound ;  there  is  a  conspicuous  total 
absence  of  higher  tones  and  all  the  lower  tones  are  weak ; 
the  fundamental  is  of  small  intensity,  containing  only  about 
4  per  cent  of  the  energy,  unless  its  pitch  lies  within  a  region 
of  characteristic  resonance.    The  vowel  ma  seems  to  have 


Fig.  165.    Characteristic  curves  for  the  distribution  of  the  energy  in  vowels  of 
Class  I,  having  a  single  region  of  resonance. 


considerable  range,  as  its  characteristic  may  vary  from  900 
to  1100;  two  curves  are  shown  for  this  vowel,  as  the  high- 
est ma  is  perhaps  the  initial  sound  from  which  all  vowels 
are  derived.  Sometimes  this  vowel  has  two  resonances  close 
together,  as  shown  in  the  curve  of  the  second  class.  The 
double  peak  for  this  vowel  is  peculiar  to  certain  voices,  and 
probably  there  is  only  one  resonance,  which  is  separated 
into  two  parts  by  the  absence  of  a  particular  partial  tone 

226 


PHYSICAL  CHARACTERISTICS 


OF  THE 


\OWELS 


from  the  sound  of  a  particular  voice ;  this  condition  is 
indicated  by  the  lower  curves  in  Fig.  163. 

The  characteristic  curves  for  the  vowels  of  the  second 
class  are  shown  in  Fig.  166,  each  having  two  characteristic 
regions  of  resonance.  The  curve  for  mot  (ma),  having  a 
single  resonance  at  the  pitch  1050,  is  placed  at  the  top, 
and  next  is  a  curve  for  the  same  vowel  in  which  there  are 


MEET 


1 

j 

s 

i 

] 

/ 

i 

>( 

\ 

12 

\ 

s 

18 

U 

6 

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19 

K 

li 

51 

3 

i 

31 

N 

1=. 

) 

EF 

G 

1 

A  B 

1, 

^3 

3 

EF 

r. 

A 

F. 

D 

EF 

..i.l. 

5 

A 

D 

EF  ( 

A 

BC^ 

) 

EF 

G  ; 

^. 

I'l 

129  259  517  1035  2069  4138 

Fig.  166.    Characteristic  curves  for  the  distribution  of  the  energy  in  vowels  of 
Class  II,  having  two  regions  of  resonance. 


two  resonances  very  close  together  at  pitches  of  950  and 
1240;  the  other  vow^els  with  the  pitches  of  their  resonance 
regions  are  :  mat,  800  and  1840  ;  met,  691  and  1953  ;  mate, 
488  and  2461  ;  and  meet,  308  and  3100.  The  lower  reso- 
nances are  practically  the  same  as  for  the  vowels  of  the  first 
class,  but  contain  only  about  50  per  cent  of  the  energy, 
while  about  25  per  cent  is  in  the  higher  region.  The  lower 
and  intermediate  tones  are  stronger  than  in  the  vowels  of 

227 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


the  first  class,  the  fundamental  often  containing  10  per 
cent  of  the  energy.  Although  each  vowel  is  characterized 
by  two  regions  of  resonance,  the  distinguishing  characteristic 
is  the  higher  resonance. 

The  characteristic  curves  show  the  resonating  properties 
of  the  vocal  cavities  when  set  for  the  production  of  the  speci- 
fied vowels,  and  they  have  true  significance  throughout 
their  lengths.  These  curves  may  be  considered  curves  of 
prohahility,  or  perhaps  they  may  be  called  curves  of  possi- 
bility, of  energy  emission  when  a  given  vowel  is  intoned. 
The  mouth  is  capable  of  selective  tone-emission  only,  that 
is,  the  only  frequencies  of  vibration  which  can  be  emitted 
at  one  time  are  in  the  harmonic  ratios.  If  the  harmonic 
scale  (see  page  169)  is  placed  upon  the  characteristic  curve 
of  a  given  vowel  with  its  first  line  at  any  designated  pitch, 
then  the  ordinates  of  the  curve  at  the  several  harmonic 
points  show  the  probable  intensities  of  the  various  partials 
when  the  particular  vowel  is  intoned  by  any  voice  at  the 
given  pitch.  These  curves  show  the  probable  intensity 
for  that  part  of  the  energy  which  is  characteristic  of  the 
vowel  in  general,  but,  since  they  are  averages  of  many 
analyses,  they  do  not  show  the  peculiarities  of  individual 
voices  aside  from  the  vowel  characteristic. 

Since  the  pitch  region  of  the  maximum  emission  of  energy 
for  a  certain  vowel  is  fixed  and  is  independent  of  the  pitch 
of  the  fundamental,  it  follows  that  the  different  vowels  can- 
not be  represented  by  characteristic  wave  forms.  When 
the  vowel  father  is  intoned  upon  the  fundamental  £2!?  = 
154,  the  sixth  partial,  6  X  154  =  924,  is  the  loudest,  and 
the  wave  form  has  six  kinks  per  wave  length.  Fig.  167,  a, 
is  an  actual  photograph  of  the  vowel  father  from  a  baritone 
voice.    When  the  same  vowel  is  intoned  by  a  soprano  voice 

228 


PHYSICAL  (  HARACTERISTICS  OF  THE  VOWELS 


at  the  pitch  B,:^  =  462,  the  second  partial,  2  X  462  =  924, 
is  the  loudest,  and  the  wave  shows  two  kinks  per  wave 
length,  as  in  h.  These  curves  are  for  the  same  vowel,  but 
are  wholly  unlike.  When  the  vowel  no  is  intoned  by  the 
baritone  at  the  pitch  B.'^^  231,  the  characteristic  is  the 


Fig.  167.    Photographs  a  and  b,  though  unUke.  are  from  the  same  vowel ;  h  and  c 
are  nearly  alike  but  are  from  different  vowels. 


second  partial,  2  X  231  =  462  ;  the  wave  has  two  kinks 
and  has  the  appearance  shown  at  c.  Wave  forms  h  and  c 
are  alike  in  general  appearance,  but  are  for  different  vowels. 
The  wave  form  therefore  depends  upon  the  pitch  of  intona- 
tion as  well  as  upon  the  vowel,  and  one  cannot  in  general 
determine  from  inspection  alone  to  what  vowel  a  given 
curve  corresponds.    Familiarity  with  the  curves  from  an 

229 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


individual  voice  will,  however,  often  enable  one  to  tell  what 
vowel  of  this  voice  is  represented. 

Either  one  of  the  word  pyramids  of  Fig.  168  forms  an 
outline  for  the  classification  of  all  vowels ;  starting  at  the 
top  and  descending  to  the  left  are  the  vowels  characterized 
by  single  resonances  of  successively  lower  pitches ;  towards 
the  right  are  those  characterized  by  two  resonances,  the 
first  of  which  descends,  while  the  second  ascends  for  the 
successive  vowels.  There  is  a  continuous  transition  from 
one  vowel  to  the  next  through  the  entire  range  of  each  class. 
The  number  of  possible  vowels  is  indefinitely  great,  having 
shades  of  tone  quahty  which  blend  one  into  another.  It 

pa  pot  ma  mot 

paw;       pat  maw  mat 

pet  met 
poe  pate  mow  mate 

pooh  peat    moo  meet 

Fig.  168.    Word  pyramids  for  classification  of  the  vowels. 

is  believed  that  any  other  vowel  from  any  language  after 
analysis  can  be  placed  upon  this  classification  frame  as 
intermediate  between  some  two  of  those  in  the  pyramid. 
It  happens  that  the  pronunciations  used  in  this  study  cor- 
respond to  nearly  uniform  distribution  of  resonances,  and 
the  vowels  are  distinct  in  sound  one  from  another ;  they 
form  what  may  therefore  be  considered  a  rational  selection 
of  standard  vowels  and  give  a  scientific  pronunciation  as  a 
basis  for  word  formation  and  for  phonetic  spelling  and 
writing. 

The  continuity  of  vowel  tone,  as  here  described,  can  be 
easily  demonstrated  by  intoning  tlie  vowel  at  the  top  of 
the  p>Tamid,  when,  without  interrupting  the  tone,  by 

230 


PHYSICAL  CHARACTERISTICS  OF  THE  \ OWELS 


gradually  closing  the  lips  one  may  cause  the  intoned 
sound  to  pass  through  all  possible  vowels  of  the  first  class, 

ma  . . .  maw  . . .  mow  . . .  moo  (pronounce  only  the  vowels, 

as,  a  ...  a  ...  0  ...  ob) .  And  again  by  starting  with  the 
same  vowel  at  the  top  of  the  pyramid,  keeping  the  lips 
in  constant  position,  but  changing  the  position  of  the 
tongue,  one  can  continuously  intone  all  the  vowels  character- 
ized by  two  resonances,  ma  . . .  mat . . .  met . . .  mate  . . .  meet 

(a ...  a  ...  e ...  a ...  e). 

Many  photographs  have  been  made  which  confirm  the 
pyramid  classification  by  simple  inspection,  showing  that 


Fig.  169.    Photographs  of  the  vowels  moo  and  meet. 


the  relations  are  based  upon  essential  features.  Compara- 
tive curves  for  the  vowels  moo  and  meet  are  given  in  Fig. 
169.  The  first  is  for  moo,  and  is  a  very  simple  curve,  the 
vowel  characteristic  being  the  single  resonance  which  is  an 
octave  higher  than  the  fundamental  and  is  represented  by 
the  wavelets  a.  The  vowel  meet  has  two  characteristic 
resonances,  the  first  of  which,  h,  is  practically  identical 

231 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


with  that  for  moo,  while  the  second  is  of  very  high  pitch 
and  is  represented  by  the  small  kinks  which  are  present 
throughout  the  curve  but  show  most  clearly  near  the  points 
b;  the  addition  of  this  high  pitch  to  the  sound  for  moo 
changes  it  to  meet. 

The  relation  of  the  vowels  moo  and  meet  is  illustrated 
by  a  common  difficulty  in  telephone  conversation.  Tele- 
phone lines  are  purposely  so  constructed  as  to  damp  out 
vibrations  of  high  frequency ;  if  the  vowel  meet  is  spoken 
into  the  transmitter,  its  high-frequency  characteristic  is  not 
carried  over  the  wire,  and  the  sound  being  heard  with  this 
part  eliminated  is  interpreted  as  moo;  for  this  reason  the 
word  three  is  often  misunderstood  as  two,  to  prevent  which 
the  r  in  three  is  trilled. 

Translation  of  Vowels  with  the  Phonograph 

The  phonograph  permits  a  simple  verification  of  the 
characteristics  of  certain  vowels,  since  the  pitch  of  the 
sounds  given  out  by  the  machine  can  be  varied  by  changing 
the  speed  of  the  motor  which  turns  the  record.  When  the 
vowel  ma  is  recorded,  the  greater  part  of  the  energy  is 
emitted  in  tones  having  a  frequency  of  about  925 ;  if  the 
record  is  reproduced  at  the  same  speed  as  that  at  which  it 
was  recorded,  one  hears  the  vowel  ma ;  but  when  the  speed 
of  rotation  is  reduced  so  that  sounds  which  previously  had 
the  pitch  925  now  have  the  pitch  735,  the  phonograph 
speaks  the  vowel  maw ;  and  still  further  reduction  of  speed 
gives  the  vowels  mow  and  moo. 

If  maw  is  recorded,  then  the  record  can  be  made  to  re- 
produce ma  by  increased  speed  of  the  motor,  and  the  other 
vowels  mow  and  moo  are  obtained  by  a  decreased  speed  as 
before. 

232 


PHYSICAT.  (IIARACTERISTICS  OF  THK  \T)WELS 


In  early  experiments  with  the  phonograph  the  vowels 
ma  and  maw  were  recorded  several  times  at  various  speeds 
of  the  cylinder,  and  afterwards  it  was  impossible  to  identify 
the  records,  because  each  could  be  made  to  reproduce  both 
vowels  perfectly. 

The  vowel  ma  was  recorded  by  the  voice  D  C  M  on  the 
phonograph,  and  without  stopping  the  cylinder,  the  phono- 
graph was  made  to  speak  this  record  into  the  phonodeik ; 
the  sound  was  photographed  and  the  speed  of  the  phono- 
graph cylinder  was  determined  at  the  same  time  ^^dth  a 
stop-watch.  Analysis  of  the  photograph  showed  that  the 
fundamental  pitch  of  intonation  was  154,  while  the  maxi- 
mum energy  of  the  sound  was  in  the  sixth  partial  tone, 
having  a  frequency  of  924  :  the  corresponding  speed  of 
the  cylinder  of  the  phonograph  was  one  turn  in  0.276 
second. 

The  speed  of  the  cylinder  was  reduced,  till  the  ear  judged 
that  the  vowel  maw  was  being  given  by  the  phonograph  ; 
the  time  for  one  turn  of  the  cylinder  was  found  to  be  0.348 
second,  corresponding  to  a  frequency  of  730  for  the  tone  of 
maximum  energy'.  Similar  trials  were  made  for  the  vowels 
mow  and  moo,  and  the  results  of  all  the  experiments  are 
shown  in  tabular  form. 


Vowel 

ma 

maw 

mow 

moo 

Speed,  sec.  per  turn 

0.276 

0.348 

0.572 

0.814 

Phonograph,  a 

924 

730 

444 

311 

Analysis,  n 

922 

732 

461 

326 

Frequencies  of  vowels  obtained  by  translation  with  the  phonograph. 


The  characteristics  from  the  phonograph  experiments 
for  ma,  maw,  mow,  and  moo,  are  924  (the  original  record), 

233 


THE  SCIENCE  OF  MUSICAE  SOUNDS 


730,  444,  and  311,  respectively,  while  the  photographic 
analyses  give  922,  732,  461,  and  326. 

Since  in  this  experiment  the  pitch  of  the  fundamental 
is  lowered  in  the  same  proportion  as  is  that  of  the  character- 
istic, to  the  abnormally  low  pitch  of  54  for  moo,  it  is  better 
to  record  the  first  tone  at  a  pitch  higher  than  that  of  normal 
speech,  or  to  make  the  record  from  a  contralto  or  soprano 
voice. 

If  the  vowel  ma  is  recorded  at  different  pitches  or  by 
different  voices,  when  the  speed  is  changed  so  that  any 
one  pitch  or  voice  gives  mow,  all  other  pitches  and  voices 
give  mow  at  the  same  time,  since  the  characteristic  for 
each  vowel  is  the  same  (approximately)  for  all  voices  and 
pitches.  A  similar  relation  exists  when  records  of  any 
one  of  the  vowels  ma,  maw,  mow,  and  moo  are  translated 
by  changed  speed  of  reproduction  to  any  other  one  of 
these  vowels. 

In  order  that  the  loudness  of  a  sound  may  remain  con- 
stant when  the  pitch  is  lowered,  the  amplitude  should 
increase,  as  was  explained  in  Lectures  II  and  V,  in  the 
proportion  shown  in  Fig.  113.  In  phonographic  reproduc- 
tion, the  amplitudes  of  the  several  component  tones  remain 
constant  as  the  frequency  is  reduced,  since  the  amphtudes 
are  determined  by  the  depth  of  the  cutting  in  the  wax ; 
this  causes  a  diminution  in  intensity  proportional  to  the 
square  of  the  speed  reduction,  and  alters  the  relative  loud- 
ness of  the  several  component  tones ;  hence  a  translated 
vowel  often  has  an  unnatural  sound,  though  it  retains  the 
vowel  characteristic.  This  difficulty  is  somewhat  over- 
come by  the  method  of  experimentation  described  below. 

When  the  vowel  moi^  is  intoned  by  a  baritone  voice  at 
the  normal  pitch  for  speech,        =  154,  the  characteristic 

234 


PHYSICAL  (  IIARA(  TERISTICS  OF  THE  VOWELS 

is  the  third  partial  of  pitch  3  X  154  =  462.  li  ma  is  re- 
corded on  the  phonograph  by  the  same  voice  at  the  same 
pitch,  the  characteristic  is  the  sixth  partial  of  pitch  6  X 
154  =  924 ;  when  the  phonograph  speed  is  reduced  to 
sound  moiv  from  this  record,  the  fundamental  pitch  becomes 
77,  and  the  characteristic  is  the  sixth  partial  of  this  lower 
pitch,  6  X  77  =  462 ;  while  this  sound  is  clearly  mow,  it  is 
not  like  a  natural  mow  of  the  baritone  voice,  being  pitched 
on  such  a  sub-bass  fundamental.  When  ma  is  recorded 
by  a  contralto  voice  on  the  fundamental  Est?  =  308,  the 
characteristic  is  the  third  partial  of  the  pitch  3  X  308  = 
924 ;  if  now  this  record  is  reproduced  at  a  slower  speed  to 
give  moiv,  the  fundamental  falls  to  E^t^  =  154,  and  the 
characteristic  is  still  the  third  partial  of  pitch  3  X  154  = 
462  ;  this  translated  ma  of  the  contralto  voice  becomes  mow 
of  the  baritone  voice  in  general  quality,  and  has  a  natural 
sound. 

Phonographic  translation  of  the  vowels  of  the  second  class, 
mat,  met,  mate,  and  meet,  is  not  possible,  for  each  has  two 
regions  of  resonance,  as  is  shown  in  Fig.  166,  the  higher  in- 
creasing in  frequency  when  the  lower  decreases. 

Whispered  Vowels 

The  vow^els  can  be  distinctly  whispered  without  the  pro- 
duction of  any  larynx  tone,  that  is,  without  fundamental 
or  pitch  and  without  the  series  of  partials  which  determine 
the  individuality  of  the  voice,  but  these  whispered  sounds 
must  contain  at  least  the  essential  characteristics  of  the 
vowels.  Photographs  of  such  whispered  vowels  are  readily 
obtained  and,  the  time  signals  being  photographed  simul- 
taneously, they  give  by  direct  measurement  the  absolute 
pitch  of  the  vowel  characteristics. 

235 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Many  photographs  have  been  taken  of  nine  vowels,  whis- 
pered by  several  voices ;  inspection  of  these  at  once  proves 
the  general  correctness  of  the  classification  of  the  vowels 
already  given. 

Whisper  records  for  the  four  vowels  of  the  first  class 
ma,  maw,  mow,  and  moo,  are  shown  in  Fig.  170,  ma  being 
at  the  top.  Since  the  usual  voice  tones  are  entirely  absent, 
each  curve  consists  mainly  of  one  frequency,  that  charac- 


FiG.  170.    Photographs  of  whispered  vowels  of  Class  I. 


teristic  of  the  vowel.  These  curves  are  arranged  in  the  or- 
der of  the  classification  and  it  is  evident  that  the  frequency 
of  the  principal  vibration  in  each  increases  from  the  lower 
to  the  upper  record. 

The  second  class  of  whispered  vowels,  mat,  m^t,  mate, 
and  meet,  is  shown  in  Fig.  171,  mat  being  at  the  top.  Each 
curve  of  this  group  has  two  distinct  frequencies ;  the  curve 
for  meet  has  the  lowest  and  the  highest,  the  high  frequency 
being  superposed  on  the  lower  like  beads  on  a  string ;  the 
other  curves  in  order  show  that  the  lower  tone  increases 

236 


PHYSICAL  CHARACTERISTICS  OF  THE  VOWELS 


in  frequency,  while  the  higher  one  decreases,  the  two  being 
quite  entangled  in  the  upper  curve  for  mat. 

The  variation  in  the  wave  form,  which  is  not  periodic 
and  therefore  is  not  caused  by  beats,  is  perhaps  due  to  the 
fact  that,  as  the  vocal  cords  are  not  in  motion,  the  vibra- 
tion of  the  air  in  the  mouth  cavity  is  uncontrolled  and  fluctu- 
ates in  both  intensity  and  pitch  within  the  characteristic 
limits. 


Fig.  171.     Phof oyiraphs  of  whispered  vowels  of  Class  II. 


Comparisons  of  forty-five  curves  of  whispers  show  fre- 
quencies for  the  characteristics  as  given  in  the  table ;  the 


Vowel 

ma 

maw 

mow 

moo 

mat 

met 

mate 

meet 

Whisper,  n 

1019 

781 

515 

383 

857 

678 

488 

391 

1890 

1942 

2385 

2915 

Analysis,  n 

922 

732 

461 

326 

800 

691 

488 

308 

1843 

1953 

2461 

3100 

Frequencies  of  the  characteristics  of  whispered  vowels. 
237 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


238 


PHYSICAL  CHARACTERISIICS 


OF  THE 


X'OWELS 


frequencies  determined  by  analysis  of  the  spoken  vowels 
are  also  given  for  reference.  It  appears  that  the  resonance 
frequency  of  the  mouth  in  whispering  is  somewhat  higher 
than  when  speaking,  though  the  whisper  characteristics 
are  well  within  the  limits  of  those  of  speech. 

Theory  of  Vowel  Quality 

The  analytical  studies  which  have  been  described  lead 
to  the  conclusion  that  intoned  vowels  are  strictly  periodic 
or  musical  sounds.  It  is  unusual,  however,  to  prolong  a 
vocal  sound  wdthout  variation ;  in  song  a  sustained  tone 
is  usually  given  some  emotional  expression,  and  in  spoken 
words  the  vowels  change  continuously  and  are  blended  mth 
the  consonants.  The  photograph  of  the  soprano  voice 
shown  in  the  frontispiece  represents  a  sustained  musical 
tone  w^hich,  though  simple  in  quality,  is  continually  chang- 
ing in  intensity.  The  flowing  of  speech  tones  from  one 
quality  to  another  is  illustrated  by  the  photographs,  Fig. 
172,  of  the  spoken  words  ''Lord  Rayleigh,"  and  Fig.  184, 
of  the  words  "  Lowell  Institute."  The  slightest  change  in 
the  sound  causes  a  change  in  the  wave  form.  It  requires 
some  practice,  but  it  is  not  impossible  to  maintain  a 
pure  vowel  tone  unchanged  for  several  seconds,  in  which 
time  there  may  be  hundreds  of  waves  which  are  truly 
periodic  ;  Fig.  173  shows  the  periodicity  of  the  vowel  sound 
mate.  Such  a  photograph  is  a  complete  justification  of  the 
application  of  harmonic  analysis  to  the  study  of  vowel  curves. 

The  mouth  with  its  adjacent  vocal  cavities  is  an  adjust- 
able resonator ;  by  varying  the  positions  of  the  jaws,  cheeks, 
tongue,  lips,  and  other  parts,  this  cavity  can  be  tuned  to  a 
large  range  of  pitches.  When  the  mouth  is  wide  open  and 
the  tongue  is  low,  the  cavity  responds  to  a  single  pitch  of 

239 


THE  SCIENCE  OF  MUSICAL  SOUNDS 

high  frequency,  and  is  set  for  the  vowel  father,  A,  Fig.  174 ; 
when  the  opening  between  the  hps  is  small,  oo,  the  pitch 
is  lowered,  as  for  gloom.  The  mouth  cavities  may  be  ad- 
justed to  reinforce  two  different  pitches  at  one  time,  as  has 
been  explained  by  Helmholtz  ;  when  set  for  the  vowel  meet, 


Fig.  173.    Photograph  of  the  vowel  mate,  showing  periodicity  of  the  wave  form. 


E,  the  cavity  responds  to  two  simple  tones,  one  correspond- 
ing to  the  back  part  of  the  oral  cavity,  and  the  other  to  the 
channel  between  the  tongue  and  the  roof  of  the  mouth. 
The  resonance  of  the  mouth  may  be  illustrated  by  holding 
forks  of  certain  pitches  before  the  mouth,  when  it  is  set 
for  the  vowel,  but  when  no  sound  is  made ;  the  fork  tone 


-•  '-g-  00 

Fig.  174.    Shape  of  mouth  cavities  when  set  for  various  vowels. 

will  be  strongly  reinforced.  Fig.  175  shows  five  forks 
corresponding  to  certain  vowel  pitches  as  determined  by 
Koenig,^^  together  with  brass  resonators  tuned  to  these 
pitches,  and  therefore  producing  the  same  effect  as  the 
mouth  when  set  for  the  several  tones. 

240 


PHYSICAL  CHARACTERISTICS  OF  THE  VOWELS 


The  variation  in  the  size  of  the  mouth  cavities,  as  between 
adults  and  children,  is  easily  compensated  by  changing 
the  opening  of  the  hps.  The  effect  of  such  a  resonator, 
having  certain  natural  periods,  can  only  be  to  modify  the 
intensity  and  phase  of  the  several  components  which  are 
already  present  in  the  sound  produced  by  the  vocal  cords ; 
the  resonator  cannot  originate  any  tone.    If  the  frequency 


Fig.  175.    Koenig's  forks  and  resonators  for  vowel  characteristics. 


of  the  resonating  cavity  coincides  with  that  of  one  of  the 
partial  tones  of  the  voice,  the  effect  must  be  to  reinforce 
this  particular  partial ;  if  the  pitch  to  which  the  mouth 
cavity  is  set  does  not  exactly  coincide  with  any  partial  of 
the  voice,  then  those  partials  whose  pitches  approximate 
that  of  the  mouth  will  still  be  favored.  Both  of  these  condi- 
tions are  illustrated  by  the  curves  of  the  eight  voices  for 
father.  Fig.  163. 

It  is  necessary  that  the  sound  generated  by  the  vocal 

R  241 


/ 


THE  SCIENCE  OF  ]\[USICAL  SOUNDS 

cords  should  be  a  composite  containing  at  least  those  par- 
tials  which  are  characteristic  of  the  vowel  to  be  spoken. 
The  sounds  of  the  voice  are  normally  very  rich  in  partials ; 
in  one  analysis  of  the  vowel  mat  was  found  every  partial 
from  one  to  twenty  inclusive ;  in  another  analysis  of  the 
same  vowel,  there  are  eighteen  partials,  the  highest  being 
number  twenty-four ;  in  the  vowel  met,  an  analysis  shows 
sixteen  partials,  the  highest  being  number  twenty-three. 

Peculiarities  of  individual  voices  are  probably  due  to  the 
presence  or  absence  of  particular  overtones  in  the  larynx 
sound,  according  to  incidental  or  accidental  conditions.  A 
low  voice  of  a  man  has  a  large  number  of  partials  not  essen- 
tial to  the  vowel,  which,  so  to  speak,  overload  the  charac- 
teristic tones ;  these  partials  may  make  the  voice  louder, 
but  they  detract  from  clearness  of  enunciation.  A  child's 
voice,  on  the  contrary,  produces  only  the  higher  tones,  and 
but  few  besides  those  necessary  for  the  vowel ;  the  enuncia- 
tion is,  therefore,  especially  clear,  clean-cut,  and  distinct. 
One  is  conscious  of  the  greater  clearness  of  enunciation  of 
a  child's  voice  when  listening  to  a  conversation  in  a  foreign 
language  which  is  understood  with  difficulty. 

The  process  of  singing  a  vowel  is  probably  as  follows. 
The  jaws,  tongue,  and  lips,  trained  by  Hfelong  practice  in 
speaking  and  singing,  are  set  in  the  definite  position  for 
the  vowel,  and  the  mouth  is  thus  tuned  unconsciously  to 
the  tones  characteristic  of  that  vowel.  At  the  same  time 
the  vocal  cords  of  the  larynx  are  brought  to  the  tension 
giving  the  desired  pitch,  automatically  if  one  is  trained  to 
sing  in  tune,  but  usually  as  the  result  of  trial.  When  the 
air  from  the  lungs  now  passes  through  the  larynx,  a  com- 
posite tone  is  generated,  consisting  of  a  fundamental  of  the 
given  pitch  accompanied  by  a  long  series,  perhaps  twenty 

242 


PHYSICAL  CHARACTERISTICS  OF  THE  VOWELS 


in  number,  of  partials,  usually  of  a  low  intensity.  The  par- 
ticular partials  in  this  series  which  are  most  nearly  in  uni- 
son with  the  vibrations  proper  to  the  air  in  the  mouth 
ca\dty,  are  greatly  strengthened  by  resonance,  and  the  result- 
ant effect  is  the  sound  which  the  ear  identifies  as  the  speci- 
fied vowel  sung  at  the  designated  pitch. 

If,  while  the  mouth  cavity  is  maintained  unchanged  in 
position,  the  vocal  cords  are  set  successively  to  different 
pitches  and  the  voice  is  produced,  then  one  definite  vow^el, 
the  same  throughout,  is  recognized  as  being  sung  at  dif- 
ferent pitches.  In  this  case  the  region  of  resonance  is  con- 
stant, though  the  pitch  of  the  fundamental  may  vary,  as 
may  also  the  pitch  and  order  of  the  particular  partials 
which  fall  within  the  region  of  resonance. 

It  follows  that  a  vowel  cannot  be  enunciated  at  a  pitch 
above  that  of  its  characteristic,  a  condition  which  is  easily 
shown  to  be  true  for  those  vowels  having  a  low-pitched 
characteristic,  such  as  gloom.  Words  which  are  sung  are 
often  difficult  to  understand ;  this  may  be  due  in  part  to 
the  fact  that  the  tones  of  the  singing  voice  are  purer  than 
those  of  speaking,  that  is,  that  they  have  fewer  partials ; 
also,  the  words  must  be  intoned  upon  pitches  assigned  by 
the  composer,  and  the  overtones  may  not  correspond  in 
pitch  with  the  characteristics  of  the  vowel ;  furthermore, 
singing  tones  are  often  too  high  to  give  the  characteristic, 
even  approximately,  as  is  explained  in  the  next  lecture. 


243 


LECTURE  VIII 


SYNTHETIC  VOWELS  AND  WORDS,  RELATIONS  OF 
THE  ART  AND  SCIENCE  OF  MUSIC 

Artificial  and  Synthetic  Vowels 

The  most  convincing  proof  of  a  vowel  theory  would  be  a 
reproduction  of  the  several  vowels  by  compounding  the 

partial  tones  ob- 
tained in  the 
analyses.  Ma- 
rage  of  Paris  has 
obtained  vowel 
sounds  by  means 
of  artificial  lar- 
ynxes  and 
mouth  and  nasal 

Fig.  176.    Apparatus  for  imitating  the  vowels.  ...  -^^ . 

cavities,  rig. 

176,  combined  with  artificial  lungs  made  of  bellows  and  an 
electric  motor.  Such  an  apparatus,  like  the  doll  that  says 
ma-ma,"  is  very  interesting,  but  it  gives  no  evidence  re- 
garding any  particular  theory  of  vowel  quality ;  the  vowels 
so  made  are  not  synthetic  reproductions  scientifically  con- 
structed, but  are  more  properly  imitations. 

Koenig  devised  the  wave  siren,  a  simple  form  of  which  is 
shown  in  Fig.  177,  for  reproducing  any  desired  wave  motion, 
the  shape  of  the  wave  being  cut  on  the  edge  of  a  disk ;  he 
also  made  a  large  wave  siren  for  compounding  sixteen  sim- 

244 


SYxNTHETIC  VOWELS 


pie  tones  of  vari- 
able loudness  and 
phase;  Fig.  178 
shows  this  appa- 
ratus.^^ These  in- 
struments have  not 
proved  successful, 
perhaps  because  the 
resulting  wave  in 
air  is  not  a  repro- 
duction of  that  cut 
on  the  disks. 

One  of  H e hn- 


FiG.  i: 


Koenif^'s  wave  siren. 


holtz's  most  celebrated  devices  is  the 
apparatus  for  reproduc- 
ing vowels  synthetically 
by   means    of  tuning 
forks. It  consists  of 
ten  electrically  driven 
tuning  forks  in  con- 
nection with 
adjustable  res- 
onators, Fig. 
179.  Helm- 
holt  z  describes 
the  experi- 
ments  by  which 
he  says  he  pro- 
duced a  very 
fine  no,  a  good 


Fig. 


Koenig' 


wave  siren  for  compounding 
partials. 


ixteen 


245 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


gloom,  and  a  passable  raw,  while  the  other  vowels  could  be 
only  imperfectly  imitated.  Zahm,  who  repeated  the  experi- 
ments with  the  Helmholtz  apparatus,  remarks  that  the 
resemblance  of  the  artificial  sounds  to  the  natural  ones  is, 
at  best,  more  or  less  fanciful.  Rayleigh  says:  ''These 
experiments  are  difficult  and  do  not  appear  to  have  been 
repeated." 

HelmhoUz  explains  the  difficulties  as  in  part  due  to  the 


fact  that  the  higher 
forks  give  only  weak 
tones  and  that  the 
series  is  not  large 
enough ;  his  highest 
fork  gave  1792  vi- 
brations per  second. 
Other  difficulties  are 
that  Helmholtz  and 
others  have  not 
known  the  exact 
composition  of  any 
of  the  vowels ;  and, 
had  the  composition 


Fig.  179.    Helmholtz's  tuning-fork  apparatus  for     been    kuOWU  there 
compounding  ten  partials.  ' 

was  no  adequate 
method  of  adjusting  the  several  components  to  the  proper 
intensity.  In  order  to  imitate  an  actual  vowel,  it  is  desir- 
able that  the  pitch  of  the  fundamental  shall  correspond 
exactly  to  that  of  the  voice  being  reproduced.  Helmholtz 
had  only  one  series  of  forks  giving  eight  partials  based  on 
112  vibrations  per  second,  which  was  later  extended  to 
give  also  eight  partials  based  on  224  vibrations. 

Helmholtz  also  tried  organ  pipes,  and  says  :  ''We  can  effect 

246 


SYNTHETIC  VOWELS 


our  purpose  tolerably  well  with  organ  pipes,  but  we  must 
have  at  least  two  sets  of  these,  loud  open  and  soft  stopped 
pipes,  because  the  strength  of  tone  cannot  be  increased  by 
additional  pressure  of  wind  without  at  the  same  time  chang- 
ing the  pitch." 

Our  study  of  organ  pipes  showed  that  these  difficulties 
are  not  insurmountable  and  that  pipes  can  be  made  which 
are  more  advantageous  than  tuning  forks  for  experiments  in 
synthesis. 

The  most  suitable  pipes  are  stopped  pipes  of  wood,  known 
as  ''Tibia"  pipes;  these  are  of  large  cross  section  in  pro- 
portion to  length,  have  narrow  mouths,  and  are  voiced  for 
low  wind  pressure.  The  pipes  have  lead  ''toes, "  the  openings 
in  which  can  be  made  larger  or  smaller,  thus  adjusting  the 
quantity  (pressure)  of  the  air  entering  the  pipes.  This 
adjustment  in  connection  with  that  of  the  lip  and  throat 
permits  any  strength  of  tone  to  be  obtained  from  the  least 
to  the  full  tone.  Every  change  in  the  strength  of  tone  causes 
a  change  in  pitch  which  must  be  compensated  by  adjusting 
the  stopper.  Many  analyses  show  that  the  tones  from  such 
pipes  have  99  per  cent  of  fundamental,  that  is,  practically, 
the  tone  is  simple. 

A  vowel  having  been  photographed  and  analyzed,  the 
synthesis  can  be  performed  in  a  strictly  quantitative  manner 
by  means  of  the  phonodeik.  A  set  of  pipes  is  prepared, 
one  pipe  for  each  partial  of  the  given  vowel ;  for  the  vowels 
as  spoken  by  D  C  M  the  least  number  of  pipes  is  six  for  the 
vowel  gloom,  while  the  vowel  father  requires  ten  pipes,  and 
the  vowel  mat  sixteen  ;  the  group  of  pipes  for  the  latter  is 
shown  in  Fig.  180. 

Even  sixteen  pipes  reproduce  only  the  more  important 
partials  of  the  vowel  mat,  since  the  full  analysis  shows  twenty 

247 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


or  more  component  tones  in  some  instances.  For  voices 
of  higher  pitches  the  number  of  pipes  required  is  less.  Each 
set  of  pipes  is  mounted  as  compactly  as  possible  on  a  separate, 


Fig.  180.    Group  of  organ  pii)es,  which,  when  sounded  simultaneously, 
reproduce  the  vowel  d  in  mat. 


small  wind  chest.  The  smaller  pipes  alternate  with  the 
larger  ones  and  their  mouths  are  on  a  different  level  to  pre- 
vent interference. 

The  analysis  of  the  voice  curve  gives  the  actual  ''ob- 

248 


synthetic:  vowt^ls 


served"  amplitudes  of  the  several  components  of  the  curve. 
The  set  of  pipes  is  now  placed  in  front  of  the  phonodeik 
in  the  position  occupied  by  the  original  voice,  and  each 
pipe  corresponding  to  a  component  tone  is  separately  ad- 
justed, till  it  shows  in  the  phonodeik  the  amplitude  required 
by  the  analysis  for  this  partial ;  the  adjustment  is  readily 
verified  as  the  amplitude  is  directly  measurable  on  the 
ground  glass  of  the  camera.  The  reproduction  is  wholly 
independent  of  the  peculiarities  of  the  phonodeik,  for  it  is 
made  with  the  same  instrument  and  under  the  same  condi- 
tions as  was  the  original  record. 

The  fundamental  pitch  is  set  from  a  piano  or  tuning  fork, 
and  the  other  partials  can  then  be  tuned  to  the  exact  har- 
monic ratios  by  means  of  the  phonodeik.  The  pipe  for  the 
second  partial,  already  in  approximate  tune,  is  sounded 
simultaneously  with  the  fundamental,  the  resulting  curve 
is  observed  on  the  ground  glass  of  the  phonodeik,  and  the 
pipe  is  tuned  until  the  wave  form  remains  constant.  The 
tuning  is  now  necessarily  perfect,  since  inexact  relationship 
produces  a  slowly  changing  curve.  The  third  and  other 
partials  are  then  successively  tuned  in  the  same  manner. 

This  method  of  tuning  is  perhaps  the  best  possible  for  two 
or  more  frequencies  which  are  in  exact  ratios,  since  it  pos- 
sesses the  advantages  of  Lissajous's  optical  method  and  is 
more  generally  applicable.  While  it  is  not  difficult  to  ad- 
just a  small  number  of  pipes  to  practically  perfect  harmonic 
frequencies,  it  is  hardly  possible  to  tune  sixteen  pipes  so  that 
the  resultant  wave  form  remains  unchanged,  and  conse- 
quently the  synthesized  tones  do  not  blend  as  perfectly  as 
do  the  partials  from  a  single  source.  With  care,  however, 
the  tuning  is  sufficiently  exact  to  secure  success  in  the  ex- 
periments. 

249 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


The  set  of  tones  thus  obtained,  when  sounded  simul- 
taneously in  front  of  the  phonodeik,  must  give  a  resultant 
wave  which  can  be  analyzed  into  the  same  components  as 
those  of  the  original  voice  curve ;  the  inference  then  is  that 
the  composite  sound  will  give  to  the  ear  the  same  vowel 
sound  as  did  the  original  voice. 


PIPES  VOICE 


Fig.  181.    Photographs  of  the  vowels  ma,  maw,  mow,  and  moo,  as  intoned  by  the 
voice  and  as  synthetically  reproduced  with  organ  pipes. 


Fig.  181  shows  several  curves  from  synthetic  pipe- vowels 
together  with  corresponding  voice  curves ;  there  is  a  close 
resemblance  between  the  two.  In  the  more  complex  pipe- 
vowels  there  is  a  continual  change  in  wave  form,  produced 
by  a  slowly  shifting  phase,  due  to  imperfect  tuning. 

In  the  oral  lectures  demonstrations  were  given  of  the  syn- 
thetic pipe-vowels  constructed  from  the  analyses  of  the 
speaker's  voice,  and  the  pipes  and  voice  were  sounded  al- 
ternately to  permit  a  direct  comparison.    There  is  no  diffi- 

250 


WORD  FORMATION 


culty  in  identifying  the  synthetic  vowels  and  in  detecting 
the  pecuHar  quahties  of  the  voice.  In  this  manner  the 
general  characteristics  of  the  vowels  father,  raw,  no,  and 
gloom,  spoken  by  D  C  M,  are  reproduced.  The  synthetic 
vowels  mat  and  hee  were  demonstrated,  the  latter  showing 
the  transformation  of  od  to  ee.  The  experiments  also 
included  the  four  different  syntheses  of  the  vowel  father, 
and  the  reproduction  of  the  simple  wwds  ma-ma  and  pa-pa, 
as  described  in  the  next  section. 

In  the  laboratory  eleven  vowels  have  been  successfully 
reproduced  by  organ-pipe  synthesis,  the  several  groups  of 
pipes  used  being  shown  in  Fig.  182.  The  same  methods 
which  make  it  possible  to  construct  the  vowels  synthetically, 
of  course,  enable  one  to  reproduce  the  tone  quality  of  any 
orchestral  instrument ;  in  this  manner  the  tones  of  the  flute, 
the  clarinet,  the  violin,  and  the  oboe  have  been  imitated. 

Word  Formation 

Experiments  with  synthetic  vowels  lead  to  interesting 
conjectures  concerning  the  origin  of  speech.  The  analyses 
of  vow^els  show^  that  the  highest-pitch,  articulate  sound 
of  the  human  voice  is  ah  as  in  father ;  if  while  intoning  this 
vowel,  the  lips  are  closed  and  opened  alternately,  the  result 
is  the  w^ord  ma-ma;  if  the  nasal  passage  is  also  closed  the 
word  is  pa-pa.  Perhaps  the  automatic  production  of  these 
sounds  by  the  infant  when  in  need  of  attention  explains 
the  origin  of  these  names  for  the  parents.  Our  organ-pipe 
talking  apparatus,  which  is,  indeed,  but  a  mere  infant,  can 
also  say  these  words. 

Referring  again  to  the  diagram  which  shows  the  analyses 
for  the  vowel  father  from  eight  voices.  Fig.  163,  we  will 
select  four,  the  voices  of  a  child,  a  boy,  a  woman,  and  a  man, 

251 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


252 


WORD  FORMATION 


for  synthetic  reproduction.  The  four  sets  of  pipes  shown  in 
Fig.  183  reproduce  this  vowel,  father,  as  intoned  by  the  four 
different  voices.  For  the  child's  high-pitched  voice  only 
three  pipes  are  required ;  when  these  are  sounded  the  vowel 
father  is  very  clearly  produced. 

If  the  supply  of  air  from  the  ''lungs"  to  the  ''throat"  is 
stopped  and  released  by  pressing  the  edge  of  the  hand  on  the 
rubber  supply  tube,  the  "infant"  is  caused  to  cry  pa-pa 


Fig.  183.    Sets  of  pipes,  reproducing  the  vowel  a  in  father,  as  intoned  Ijy  four 

different  voices. 


with  a  voice  that  would  be  considered  human,  if  the  source 
were  unknown.  Similarly  the  pipes  for  the  contralto  voice, 
six  in  number,  say  pa-pa  perhaps  even  more  naturally, 
while  the  boy's  voice,  eight  pipes,  and  the  baritone,  ten 
pipes,  speak  the  same  word  very  clearly ;  the  latter  being 
the  speaker's  voice,  the  word  was  actually  spoken  for  com- 
parison. By  thus  using  the  vowels  in  a  word,  the  success 
of  the  synthetic  reproduction  is  the  more  easily  recognized. 

253 


THE  SCIENCE 


OF  MUSICAL  SOUNDS 


254 


WORD  FORMATION 


While  the  four  sets  of  pipes  are  entirely  different,  contain- 
ing from  three  to  ten  pipes  each,  and  while  the  general  tone 
qualities  are  very  different,  yet  each  contains  the  same  vowel 
characteristic  as  its  loudest  component,  and  each  distinctly 
sounds  the  vowel  father.  The  comparison  of  the  four  is  a 
striking  illustration  of  the  truthfulness  of  the  analyses. 

If  the  flow  of  air  is  only  partially  interrupted,  the  sound 
does  not  altogether  cease  between  syllables,  and  the  pipes 
pronounce  ma-ma;  this  is  better  shown  by  alternately 
producing  pa-pa  and  ma-ma. 

If  the  same  experiment  is  made  upon  the  synthetic  vowel 
raw,  the  word  paw-paw  is  spoken,  and  if  the  vowel  mat  is 
used,  the  word  pd-pd  is  obtained ;  thus  we  have  the  three 
pronunciations,  pa-pa,  paw-paw,  and  pd-pd. 

It  is  certainly  possible,  by  further  experimentation,  to 
produce  various  other  noises,  explosive  effects,  hisses,  etc., 
which  when  combined  with  the  vowel  tones  already  produced 
will  form  other  words,  as  sss-ee,  bb-ee,  a-tt,  nn-o,  etc. 

In  addition  to  the  imperfect  tuning  the  absence  of  in- 
flection and  tone  flow  gives  an  unnatural  effect  to  the  syn- 
thetic words.  In  speaking  the  word  pa-pa,  for  instance, 
the  two  syllables  are  usually  uttered  at  different  pitches, 
and  in  any  phrase  there  is  an  almost  continual  change  of 
quality  and  quantity  of  tone,  as  is  shown  in  Fig.  184,  which 
is  a  photograph  of  the  words  Lowell  Institute"  as  spoken 
by  D  C  M.  These  words  were  spoken  in  a  natural  manner, 
in  about  one  and  two  tenths  seconds.  The  first  word  and 
part  of  the  second,  represented  by  the  three  syUables  low-ell- 
ins,  consist  of  a  continuous  flow  of  sound,  a  kind  of  mel- 
ody of  vowel  tone."  In  contrast,  the  word  ins-ti-tute  is 
disrupted  by  the  fs  which  consist  of  interruptions  of  the 
sound  lasting  about  a  tenth  of  a  second  each. 

255 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


The  exact  form  of  the  record  depends  not  only  upon  the 
words  spoken,  but  also  quite  as  much  upon  the  pecuharities 
of  the  individual  voice  and  upon  the  resonance  characteris- 
tics of  the  recording  apparatus.  The  incidental  variations 
and  the  extensive  vocabulary  of  language,  would  produce 
an  enormously  extended  set  of  records,  yet  it  is  possible  to 
learn  to  ''read"  the  sound-wave  records  of  spoken  words; 
such  records  constitute  a  system  of  phonetic  writing,  but 
the  system  is  neither  ''shorthand"  nor  "simplified  spelling." 

Such  records  are,  doubtless,  capable  of  being  transformed 
into  corresponding  movements  of  the  air,  and  thus  the  orig- 
inal sound  may  be  reproduced  after  the  manner  of  the 
talking  machine. 

These  experiments  suggest  a  scientific  method  of  word 
formation.  A  series  of  eight  distinctly  different  vowels 
has  been  established  by  quantitative  analysis ;  these  vowels 
may  be  combined  with  the  various  consonant,  explosive, 
and  aspirant  sounds  according  to  a  systematic  scheme, 
forming  sets  of  words  each  having  a  distinct  pronunciation. 
Any  word  formula  having  been  arbitrarily  selected,  a  group 
of  eight  words  may  be  formed,  as  shown  in  the  following 
table.  If  the  sound  represented  by  the  letter  m  is  placed 
before  each  vowel,  we  have  the  words  in  the  first  column ; 
words  thus  formed,  which  have  not  been  in  use  in  the  English 
language,  are  printed  in  italics.  If  to  each  of  these  words 
is  added  a  final  s,  the  words  of  the  second  column  are  ob- 
tained, while  the  addition  of  a  t  gives  the  words  in  the  third 
column.  If  the  word  formula  chosen  is  that  for  bottle, 
the  eight  words  of  the  sixth  column  are  formed,  only  three 
of  which  are  in  use.  Other  combinations  are  shown  in  the 
table. 

Such  a  scheme  of  word  formation  contains  interesting 

256 


WORD  FORMATION 


suggestions  for  uniform  and  simplified  spelling,  and  also  for  a 
uniform  pronunciation. 


I 

ma 

mas 

mot  ^ 

baa 

Kr>Vi 

Daii 

bot 

bottle 

II 

maw 

moss 

maught  < 

^ba 
baw 

bought 

bawtle 

III 

mow- 

mose 

moat 
\  mote 

,'  beau 
bow 

,  boat 
\bot 

boatle 

IV 

moo 

moos 

moot 

booh 

boot 

bootle 

V 

ma 

■  mass 
^  mas 

mat 

ba 

bat 

battle 

V  I 

me 

mess 

met 

be 

bet 

oettie 

VII 

may 

mace 

mate 

bay 

bate 
■  bait 

baitle 

VIII 

me 

'meese 
\  mease 

;  meet 
\  meat 

be 
,bee 

j  beet 
\beat 

beetle 

I 

pa 

pot 
^  pawt 

ha 

hot 

hock 

rah 

II 

paw 

paut 

haw 

haught 

hawk 

raw 

III 

poe 

pote 

/ho 
hoe 

hate 

hoke 

/  roe 
1^  row 

IV 

pooh 

pool 

/  hoo 
\who 

hoot 

hook? 

rue 

V 

pa 

pat 

ha 

hat 

hack 

ra 

VI 

pe 

pet 

he 

het 

heck 

re 

VII 

pay 

pate 

hay 

hate 

hake 

/ray 
\re 

VIII 

pea 

peat 

he 

heat 

heek 

ree 

foat 
foot .? 
fat 
fet 

fate 

/feet 
\feat 

/roc 

\rock 

ra  wk 
roak 

rook  f 

/rack 
1  wrack 
J  reck 
\  wreck 

rake 

/reek 
\ wTeak 


tot 

/taut 
\  taught 

tote 
toot 
tat 

tet 

/tate 
\tait 

teat 


Vocal  and  Instrumental  Tones 

The  difference  between  vocal  and  instrumental  tones  is 
exhibited  by  the  analyses  shown  in  Fig.  185,  which  illus- 
trates the  fixed-pitch  theory  and  the  relative-pitch  theory  of 
tone  quality.  The  upper  lines  of  the  figure  show  the  relative 
loudness  of  the  several  partials  for  the  vowel  father,  intoned 
at  three  different  pitches ;  the  maximum  resonance  has  a 
s  257 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


fixed  pitch  of  about  k^^=  922,  the  loudest  partial  being  in 
turn  the  sixth,  the  fifth,  and  the  fourth  as  the  pitch  of  the 
fundamental  rises.  The  lower  lines  of  the  figure  show  flute 
tones  at  three  different  pitches.  The  loudest  component 
is  always  the  second  partial,  which  changes  pitch  with  the 
fundamental ;  thus  it  is  the  relation  of  partials  that  is  con- 
stant, and  not  the  pitch  of  the  partial. 


Co  D  EF 


GABC,  DEFGA 


D    EF    G    A    BCg  D    EF    G    A    BCfi  D    EF    G   A  BC7 

'    '  '    '    '    '  I  .  I  .  I  I  .  I  .  I  .  I  r.,l,.,l.l..,l..,i..,i. 


|j|j|§l?|i|i|j|j|j|t|g|  lilgliliiHiM 


mm 


s  s  §  s  H  sa 


129 


259  517  1035  2069 

Fig.  185.    Analyses  of  voice  and  flute  tones. 


4138 


It  may  be  asked  what  is  the  effect  of  raising  the  pitch  of 
the  flute  tone  a  little  higher  than  shown,  till  its  maximum 
loudness  agrees  with  that  of  the  vowel  father.  When  the 
flute  sounds  Asjj  =  461,  since  the  second  partial  is  the 
loudest,  the  maximum  energy  is  at  A^Jf  =  922,  and  the  tone 
actually  has  a  resemblance  to  the  vowel.  By  stopping  the 
breath,  somewhat  as  is  done  in  speaking  the  word  pa-pa,  the 
flute  imitates  the  vowel. 

258 


WORDS  AND  MUSIC 


Opera  in  English" 

The  characteristics  of  the  several  vowels,  which  were 
described  in  the  previous  lecture,  are  shown,  superposed, 
in  Fig.  186.  The  essential  part  of  each  vowel  is  a  component 
tone  or  tones,  the  pitch  of  which  is  within  certain  limits  in- 
dicated by  the  corresponding  curve.    A  characteristic,  the 


C2  D   EF    G   A    BC3  D   EF    G   A    BC4   D   EF    0   A    BCg  D    E  F    G    A    B  Cg  D    E  F    G   A  BC7 

129  259  517  1035  2069  4138 

Fig.  186.    Superposed  curves  of  vowel  characteristics  showing  relations  to  voice 

ranges. 

pitch  of  which  comes  in  the  region  where  the  curves  for 
two  vowels  overlap,  would  produce  an  intermediate  vowel 
which  might  serve  for  either  of  those  specified,  and,  although 
the  pronunciation  of  such  a  vowel  might  be  different  from 
that  used  in  defining  the  characteristics,  it  would  be  readily 
interpreted  when  used  in  a  w^ord.  This  shaded  pronuncia- 
tion of  a  vowel  would  be  accepted  in  singing,  where  perfect 
enunciation  is  not  expected.    Many  singing  tones  have 

259 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


pitches  above  the  characteristic  ranges  of  certain  vowels, 
such  as  gloom  and  meet,  and  these  vowels  cannot  be  sung 
properly  at  the  higher  pitches.  The  vowel  father  has  a 
characteristic  higher  than  any  singing  tone  and  can  be  sung 
under  any  circumstances  of  voice  or  pitch.  Thus  there  is  a 
scientific  reason  for  the  free  use  of  such  syllables  as  tra-la-la 
in  vocal  exercises.  Yodeling  is  probably  the  easy  flowing  of 
varying  vowel  tones  to  fit  the  melody. 

The  characteristics  of  the  several  vowels  are  given  in  musi- 
cal notation  in  Fig.  187  ;  the  notes  correspond  to  the  pitches 


^  *- 
#a  ^  ^  =  = 




moo  mow     maw    ma   mot  mat  met  mate  meet 
pooh  poe      paw    pa   pot  pat  pet  pate  peat 

Fig.  187.    Characteristics  of  the  vowels  in  musical  notation. 

of  maximum  resonance,  as  shown  by  the  curves  of  the  pre- 
vious figure,  and  each  vowel  can  be  most  clearly  intoned  upon 
the  corresponding  note.  A  vowel  can  also  be  freely  intoned 
upon  any  lower  note  of  which  the  characteristic  note  is 
a  harmonic,  such  as  notes  an  octave,  a  twelfth,  or  a  fifteenth 
lower  (in  musical  intervals)  than  the  characteristic.  A 
baritone  voice  can  easily  intone  the  vowel  gloom,  in  falsetto, 
upon  the  characteristic  pitch  of  E3,  shown  in  the  figure. 
The  characteristics  of  all  the  vowels  can  be  verified  by  the 
test  of  free  enunciation. 

A  consideration  of  the  characteristics  of  the  vowels  leads 
to  certain  definite  conclusions  regarding  the  question,  so 

260 


WORDS  AND  MUSIC 

widely  discussed,  whether  grand  opera  originally  written  in 
a  foreign  language  should  be  sung  in  EngHsh.  No  doubt 
every  composer  sets  words  to  music  with  some  regard  for 
effective  rendition,  in  doing  which  he  conforms,  perhaps 
unconsciously,  to  the  natural  requirements.  Suppose  that 
in  the  original  the  composer  set  the  vowel  raw,  having  a 
characteristic  pitch  of  about  732,  to  the  melody  note  F4  ^,  of 
the  same  pitch,  the  vowel  can  then  be  sung  and  enunciated 
with  ease.  If,  in  the  translation,  some  other  vowel,  as  no, 
the  characteristic  pitch  of  which  is  461,  falls  upon  this 
note  its  proper  enunciation  will  be  difficult,  or  impossible, 
since  it  must  be  sung  at  the  pitch  732.  The  vocalist  in 
attempting  to  sing  the  vowel  will  find  the  result  vocally 
deficient  and  the  effort  perhaps  physically  painful,  and  will 
be  emphatically  of  the  opinion  that  translated  opera  is  im- 
practicable. Furthermore,  the  auditors  will  hardly  under- 
stand the  English  words  with  the  forced  and  imperfect 
vowels  any  better  than  they  understood  the  foreign  language. 
If  the  translator  arranges  the  vowels  upon  the  same  notes 
as  were  used  in  the  original,  or  upon  others  equally  suitable, 
the  translated  opera,  so  far  as  this  element  goes,  will  be  just 
as  satisfactory  to  both  the  vocalist  and  the  auditor  as  was  the 
original.  The  effectiveness  of  vocal  music  is  not  dependent 
upon  the  nationality  of  its  words,  but  upon  the  suitability 
of  melody  to  vowels,  a  condition  which  the  composer  fulfills 
through  his  artistic  instinct.  The  translator  of  an  opera 
must  secure  this  adaptation  by  his  skill ;  he  needs  to  be 
not  only  a  linguist  and  a  poet,  but  also  a  musician  and 
even  somewhat  of  a  physicist,  since  he  must  constantly  be 
guided  by  the  facts  represented  in  the  curves  of  vowel 
characteristics ;  such  a  combination  of  artist  and  scientist 
is  very  rare. 

261 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


It  has  been  suggested  that  certain  songs  and  choruses 
which  are  especially  effective  owe  this  quaUty  to  the  proper 
relation  of  vowel  sounds  to  melody  notes,  and  the  Hallelu- 
jah Chorus  from  the  ''Messiah"  has  been  cited  as  an 
instance. 

When  one  considers  the  authors  of  the  arguments  which 
have  been  pubUshed  concerning  translated  opera,  it  will  be 
found  that  some  soprano  singers  are  opposed  to  the  transla- 
tion, while  many  of  those  who  favor  it  are  baritones.  The 
lines  at  the  top  of  the  diagram.  Fig.  186,  indicate  the  ranges 
in  pitch  of  soprano  and  bass  voices.  The  pitch  of  the  so- 
prano voice  in  singing  often  rises  above  that  of  the  charac- 
teristics of  certain  vowels,  which  then  become  difficult ; 
the  bass  voice  when  highest  is  still  below  the  lowest  vowel 
characteristic,  and  it  is  thus  able  to  intone  any  vowel  on 
any  note  which  it  can  sing ;  to  one,  translated  opera  seems 
ineffective,  while  to  the  other  it  causes  no  difficulty. 

Relations  of  the  Art  and  Science  of  Music 

In  the  lectures  now  brought  to  a  close  we  have  very  briefly 
explained  the  Science  of  Musical  Sounds  and  have  incom- 
pletely described  some  of  the  methods  and  results  of  Sound 
Analysis.  The  science  of  sound  is  related  to  at  least  three 
phases  of  human  endeavor,  the  intellectual,  the  utilitarian, 
and  the  aesthetic.  In  conclusion  we  will  refer  to  some 
of  these  relations  without  extended  comment  since  an 
adequate  discussion  would  require  several  lectures. 

The  appreciation  of  knowledge  for  its  own  sake  is  general ; 
and  what  knowledge  should  be  more  valued  than  that  con- 
cerning sound,  related  as  it  is  to  many  of  the  necessities 
as  well  as  to  the  luxuries  of  existence?  It  is  true  of  the 
science  of  sound,  as  well  as  of  all  others,  that 

262 


THE  SCIENCE  AND  ART  OF  MUSIC 


"The  larger  grows  the  sphere  of  knowledge 
The  greater  becomes  its  contact  with  the  unknown." 

Helmholtz,  Koenig,  and  Rayleigh,  by  observation,  experi- 
ment, and  theory,  have  developed  this  science  to  magnifi- 
cent proportions,  yet  the  realm  of  nature  is  so  vast  and 
varied  that  some  other  indefatigable  discoverer  may  be  able 
to  push  forward  into  unknown  regions,  and  climbing  the 
height  of  some  discovery,  see  the  inspiring  prospect  of  new, 
though  far  distant,  truths,  and  be  thrilled  with  the  desire  for 
their  possession.  The  challenge  of  the  unknown  and  the 
joy  of  discovery  inspire  him  to  devote  himself  to  further 
exploration. 

While  formerly  the  regard  for  science  was  largely  confined 
to  the  academic  world,  within  the  last  few  years  there  has 
arisen  a  remarkable  and  widespread  appreciation  of  scien- 
tific methods,  and  now  industrial  and  commercial  enterprises 
are  appealing  to  the  scientist  for  assistance.  No  sooner  is  a 
new  scientific  fact  or  process  announced  than  there  is  an 
inquiry  as  to  its  usefulness.  The  science  of  sound  will 
be  found  ready  to  satisfy  the  utilitarian  demands  which 
will  be  made  upon  it. 

The  science  of  sound  should  be  of  inestimable  benefit 
in  the  design  and  construction  of  musical  instruments,  and 
yet  with  the  exception  of  the  important  but  small  work  of 
Boehm  in  connection  with  the  flute,  science  has  not  been 
extensively  employed  in  the  design  of  any  instrument. 
This  can  hardly  be  due  to  the  impossibility  of  such  applica- 
tion, but  rather  to  the  fact  that  musical  instruments  have 
been  mechanically  developed  from  the  vague  ideas  of  the 
artist  as  to  the  conditions  to  be  fulfilled.  When  the  artist, 
the  artisan,  and  the  scientist  shall  all  work  together  in 
unity  of  purposes  and  resources,  then  unsuspected  develop- 

263 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


merits  and  perfections  will  be  realized.  These  possibilities 
are  becoming  manifest  in  relation  to  the  piano,  the  organ, 
and  some  other  instruments. 

The  artistic  and  aesthetic  musician  has  been  wont  to  dis- 
parage, if  not  ridicule,  the  development  of  mechanical 
musical  instruments,  such  as  the  player-piano,  yet  I  believe 
that,  since  the  establishment  of  the  equally  tempered 
musical  scale  by  Bach,  nothing  else  will  have  contributed 
so  much  to  the  aesthetic  development  of  musical  art.  The 
inventor  of  the  Pianola  little  dreamed  that  the  mechanical 
operation  of  the  piano  would  lead  to  a  thorough  scientific 
study  of  music  and  musical  instruments,  but  such  has  been 
the  result.  The  simple  mechanical  player  has  been  de- 
veloped into  elaborate  devices  for  the  complete  reproduction 
of  artistic  performances.  The  success  of  these  synthetic 
musical  instruments  depends  upon  an  analytical  knowledge 
of  all  the  factors  of  sound  and  music ;  that  is,  the  pure 
science  of  these  subjects  must  be  brought  to  bear  upon 
the  practical  problems  involved.  The  mechanical  precision 
of  such  instruments  reacts  critically  upon  the  artist-per- 
former and  the  composer,  resulting  in  greater  artistic  per- 
fection ;  the  unhmited  technical  possibilities  of  the  machine 
is  an  incentive  to  the  composer  to  write  music  with  greater 
freedom. 

The  marvelous  inventions  of  the  telephone  and  the  talking 
machine  could  never  have  been  developed  without  the  aid 
of  pure  science ;  a  knowledge  of  the  science  of  electricity 
and  magnetism  and  of  mechanism  is  not  sufficient  for  their 
perfection ;  an  increased  knowledge  of  the  science  of  sound 
is  also  required. 

The  utilitarian  application  of  the  science  of  sound  is 
nowhere  better  illustrated  than  in  the  design  of  auditoriums ; 

264 


THE  SCIENCE  AND  ART  OF  MUSIC 


for  of  what  avail  is  a  perfected  musical  instrument  con- 
trolled by  a  master,  or  of  what  effect  is  an  oration  pronounced 
with  faultless  elocution,  if  the  auditors  are  placed  in  sur- 
roundings which  distort  and  confuse  the  sound  waves  so 
that  intelligent  perception  is  impossible? 

The  artistic  world  has  rather  disdainfully  held  aloof 
from  systematic  knowledge  and  quantitative  and  formu- 
lated information ;  this  is  true  even  of  musicians  whose  art 
is  largely  intellectual  in  its  appeal.  The  student  of  music 
is  rarely  given  instruction  in  those  scientific  principles  of 
music  which  are  established.  Years  are  spent  in  slavish 
practice  in  the  effort  to  imitate  a  teacher,  and  the  mental 
faculties  are  driven  to  exhaustion  in  learning  dogmatic 
rules  and  facts.  Bach  said  music  is  the  greatest  of  all 
sciences'' ;  while  this  comparison  may  not  be  true  at  the 
present  time,  yet  the  construction  of  the  equally  tempered 
scale  is  clearly  scientific,  and  it  is  no  doubt  true  that  the 
relations  of  the  major  and  the  minor  scales,  and  the  nature 
of  chords  and  their  various  forms  and  progressions,  as  well 
as  many  other  fundamental  principles,  can  be  explained 
better  by  science  than  by  precept.  Experience  indicates 
that  a  month  devoted  to  a  study  of  the  science  of  scales 
and  chords  and  of  melody  and  harmony,  will  advance  the 
pupil  more  than  a  year  spent  in  the  study  of  harmony  as 
ordinarily  presented. 

Regarding  the  art  of  music  Helmholtz  says :  ''Music  was 
forced  first  to  select  artistically,  then  to  shape  for  itself, 
the  material  on  which  it  works.  Painting  and  sculpture  find 
the  fundamental  character  of  their  materials,  form  and  color, 
in  nature  itself,  which  they  strive  to  imitate.  Poetry  finds 
its  material  ready  formed  in  the  words  of  language.  Archi- 
tecture has,  indeed,  also  to  create  its  own  forms ;  but  they 

265 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


are  partly  forced  upon  it  by  technical  and  not  by  purely  artis- 
tic considerations.  Music  alone  finds  an  infinitely  rich  but 
totally  shapeless  plastic  material  in  the  tones  of  the  human 
voice  and  artificial  musical  instruments,  which  must  be 
shaped  on  purely  artistic  principles,  unfettered  by  any 
reference  to  utility  as  in  architecture,  or  to  the  imitation  of 
nature  as  in  painting,  or  to  the  existing  symbolical  meaning 
of  sounds  as  in  poetry.  There  is  a  greater  and  more  absolute 
freedom  in  the  use  of  the  material  for  music  than  for  any 
other  of  the  arts.  But  certainly  it  is  more  difficult  to  make 
a  proper  use  of  absolute  freedom,  than  to  advance  where 
external  irremovable  landmarks  limit  the  width  of  the 
path  which  the  artist  has  to  traverse.  Hence  also  the  culti- 
vation of  the  tonal  material  of  music  has,  as  we  have  seen, 
proceeded  much  more  slowly  than  the  development  of  the 
other  arts." 

In  Music  and  the  Higher  Education,"  Professor  Dickin- 
son says :  ''Strange  as  it  may  seem  that  notes  'jangled,  out 
of  tune  and  harsh,'  should  give  pleasure  to  any  one  of  average 
intelligence,  yet  the  abundance  of  evidence  that  they  do  so 
indicates  that  the  training  of  the  youthful  ear  to  discrimina- 
tion between  the  pure  and  the  impure  is  not  to  be  neglected. 
.  .  .  The  guide  to  musical  appreciation  need  not  deem  his 
effort  wasted  when  he  preaches  upon  the  need  of  preparing 
the  auditory  sense  to  catch  the  finer  shades  of  tone  values. 
.  .  .  Let  the  music  lover  not  be  content  with  imperfect  in- 
tonation, let  him  learn  to  detect  all  the  shades  of  timbre  which 
instruments  and  voices  afford,  let  him  train  himself  to  per- 
ceive the  multitudinous  varieties  and  contrasts  which  are 
due  to  the  relative  prominence  of  overtones  .  .  .  and  while 
his  ear  is  invaded  by  the  surge  and  thunder  of  the  full  or- 
chestra, let  him  try  to  analyze  the  thick  and  luscious  current 

266 


THE  SCIENCE  AND  ART  OF  MUSIC 

into  its"  elements,  .  .  .  turning  the  dense  mass  of  tone  color 
into  a  huge  spectrum  of  scintillating  hues!" 

For  the  fullest  accomplishment  of  these  ends  the  musician 
may  well  appeal  to  science.  Such  mathematical  and  physi- 
cal studies  as  we  have  described  prepare  the  'infinitely 
rich,  plastic  material"  out  of  which  music  is  made,  and  they 
provide  the  methods  and  instruments  for  its  analytical 
investigation.  Nevertheless,  that  which  converts  sound 
into  a  grand  symphony  and  exalts  it  above  the  experiments 
of  the  laboratory  is  something  free  and  unconstrained  and 
which  therefore  cannot  be  expressed  by  a  formula.  The 
creations  of  fancy  and  musical  inspiration  cannot  be  made 
according  to  rule,  nor  can  they  be  made  upon  command. 

Wagner,  one  of  the  most  inspired  musicians  that  the  world 
has  ever  known,  was  offered  a  great  sum  of  money  by  the 
Centennial  Commission  to  compose  a  Grand  March  for  the 
opening  exercises  of  the  Philadelphia  Exposition  of  1876. 
Under  the  base  influence  of  mere  gold  which  he  needed  to 
pay  debts,  he  wrote  a  very  ordinary  piece  of  music,  quite 
unworthy  of  himself,  and  of  which  he  was  ashamed.  In 
contrast  to  this,  while  under  the  inspiration  of  the  death  of 
a  mythical  hero  in  one  of  his  great  music  dramas,  Wagner 
wrote  the  Siegfried  Death  Music,  sometimes  called  the 
Funeral  March,  which  as  performed  at  Bayreuth  is  perhaps 
the  greatest  and  most  sublime  piece  of  instrumental  music 
ever  heard  by  man ;  it  is  a  most  profound  expression  of 
abstract  grief. 

In  ''The  Mysticism  of  Music  "  the  late  R.  Heber  Newton 
says :  ''Our  modern  world  is  not  more  distinctively  the  age 
of  science  than  it  is  the  age  of  music ;  music  is  the  art  of  the 
age  of  knowledge.  Music  is  an  emotional  symbolism,  sug- 
gesting that  which,  as  feehng,  Hes  beyond  all  words  and 

267 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


thoughts.  'Where  words  end,  there  music  begins.'  Music 
can  never  cease  to  be  emotional,  because  thought,  in  pro- 
portion as  it  is  deep  and  earnest,  always  trembles  into 
feelings.  The  most  holy  place  in  the  universe  is  the  soul 
of  man.  All  sciences  lead  us  up  to  the  threshold  of  this  inner 
creation,  this  unseen  universe,  throw  the  door  ajar  and  point 
us  within.  All  arts  pass  through  the  open  door  into  the 
vestibule  of  this  inner  temple.  Music  takes  us  by  the  hand, 
boldly  leads  us  within,  and  closes  the  door  behind  us." 
(These  sentences  have  been  selected  from  the  first  twenty- 
nine  pages.) 

Music  is  indeed  a  mysterious  phenomenon  ;  tones,  noises, 
rhythms,  time  values,  mathematical  ratios,  and  even 
silences,  all  conveyed  to  the  ear  by  mere  variations  in  air 
pressure,  are  its  only  means  of  action,  yet  with  these  it 
awakens  the  deepest  emotions. 

Du  Maurier,  in  ''Peter  Ibbetsen,"  describes  the  impres- 
siveness  of  musical  sounds  as  follows:  "The  hardened  soul 
melts  at  the  tones  of  the  singer,  at  the  unspeakable  pathos 
of  the  sounds  that  cannot  lie ;  ...  one  whose  heart,  so 
hopelessly  impervious  to  the  written  word,  so  helplessly 
callous  to  the  spoken  message,  can  be  reached  only  by  the 
organized  vibrations  of  a  trained  larynx,  a  metal  pipe,  a 
reed,  a  fiddle  string  —  by  invisible,  impalpable,  incompre- 
hensible little  air- waves  in  mathematical  combinations, 
that  beat  against  a  tiny  drum  at  the  back  of  one's  ear. 
And  these  mathematical  combinations  and  the  laws  that 
govern  them  have  existed  forever,  before  Moses,  before 
Pan,  long  before  either  a  larynx  or  a  tympanum  had  been 
evolved.    They  are  absolute!  " 

I  would  like  to  quote  again  from  the  ''Letters  of  Sidney 
Lanier,"   the  poet-musician.    "  'Twas  opening  night  of 

268 


THE  SCIENCE  AND  ART  OF  MUSIC 


Thomas'  orchestra,  and  I  could  not  resist  the  temptation  to 
go  and  bathe  in  the  sweet  amber  seas  of  this  fine  music,  and 
so  I  went,  and  tugged  me  through  a  vast  crowd,  and  after 
standing  some  while,  found  a  seat,  and  the  baton  tapped  and 
waved,  and  I  pUmged  into  the  sea,  and  lay  and  floated.  Ah  ! 
the  dear  flutes  and  oboes  and  horns  drifted  me  hither  and 
thither,  and  the  great  \dolins  and  small  violins  swayed  me 
upon  waves,  and  overflowed  me  with  strong  lavations,  and 
sprinkled  gUstening  foam  in  jm}^  face,  and  in  among  the 
clarinetti,  as  among  waving  water-lilies  vdth  flexible  stems, 
I  pushed  my  easy  way,  and  so,  even  lying  in  the  music- waters, 
I  floated  and  flowed,  my  soul  utterly  bent  and  prostrate." 

And  again  Lanier  writes  of  an  orchestral  performance, 
in  which  he  himself  was  the  principal  flutist  :  '^Then  came 
our  piece  de  resistance,  the  '  Dream  of  Christmas '  overture,  by 
Ferdinand  Hiller.  Sweet  Heaven  — •  how  shall  I  tell  the 
gentle  melodies,  the  gracious  surprises,  the  frosty  glitter  of 
star-light,  and  flashing  of  icy  spicules  and  of  frozen  surfaces, 
the  heart}"  chanting  of  peace  and  good-will  to  men,  the 
thrilling  pathos  of  virginal  thoughts  and  trembling  anticipa- 
tions and  lofty  prophecies,  the  solemn  and  tender  breathings- 
about  of  the  coming  reign  of  forgiveness  and  of  love,  and 
the  final  confusion  of  innumerable  angels  fl^^ng  through  the 
heavens  and  jubilantly  choiring  together." 

A  musician  must  be  skilled  in  the  technic  of  music,  he 
must  be  trained  in  musical  lore,  and  above  all  else  he  must 
be  an  artist ;  when  to  these  qualifications  is  added  inspira- 
tion, the  conditions  are  pro\ided  which  have  given  to  the 
world  its  greatest  musicians.  But  will  not  the  creative  mu- 
sician be  a  more  powerful  master  if  he  is  also  informed  in 
regard  to  the  pure  science  of  the  methods  and  materials 
of  his  art  ?    Will  he  not  be  able  to  mix  tone  colors  with 

269 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


greater  skill  if  he  understands  the  nature  of  the  ingredients 
and  the  effects  which  they  produce?  Does  not  the  inter- 
pretative musician  need  a  knowledge  of  the  capacity,  possi- 
bilities, and  limitations  of  the  tonal  facilities  at  his  command, 
as  well  as  a  knowledge  of  the  construction  of  the  written 
music,  in  order  that  he  may  render  a  composition  with 
consummate  effect?  While  the  science  of  music,  because 
of  its  incomplete  development,  has  never  exerted  its  full 
influence  on  the  art,  yet  it  should  be  appreciated  and  more 
generously  cultivated  for  the  great  assistance  which  it  can 
be  made  to  yield. 

Not  only  will  the  creative  and  interpretative  artist  be  the 
better  able  to  control  the  purely  mechanical  means  of 
operation  because  of  complete  knowledge,  but  the  receptive 
musician  will  derive  greater  pleasure  from  this  physical 
phenomenon  if  he  is  also  cognizant  of  its  marvelous  but 
systematic  complexity.  The  musically  uncultivated  and 
scientifically  untrained  listener  may  greatly  enjoy  music, 
but  this  enjoyment  is  a  gratification  of  the  senses.  If  to 
this  pleasure  of  sensation  is  added  the  intellectual  satisfac- 
tion of  an  understanding  of  the  purposes  of  the  composer, 
the  facilities  at  the  command  of  the  interpreter,  and  the 
physical  effects  received  by  the  hearer,  then  music  truly 
becomes  a  source  of  exquisite  delight  which  so  pervades 
and  thrills  one's  being  that  he  is  carried  away  ''on  the  golden 
tides  of  music's  sea."  No  other  art  than  music,  through 
any  sense,  can  so  transport  one's  whole  consciousness  with 
such  exalted  and  noble  emotions. 


270 


APPENDIX 


REFERENCES 

General  References  : 

H.  von  Helmholtz,  Sensations  of  Tone,  translated  by  A.  J.  Ellis,  2 
English  ed.,  London  (1885),  576  pages.  The  most  complete  account  of 
the  phenomena  of  sound  as  related  to  sensation  and  to  music. 

H.  von  Helmholtz,  Vorlesungen  ilber  die  mathematischen  Principien  der 
Akustik.  Liepzig  (1898),  256  pages.  A  concise  mathematical  treat- 
ment of  certain  acoustic  phenomena,  as  treated  by  Helmholtz  in  his 
university  lectures. 

Lord  Rayleigh,  Theonj  of  Sound,  2  vols.,  2  ed.,  London  (1894),  480  +  504 
pages.  The  most  comprehensive  treatise  on  the  theory  of  sound,  largely 
mathematical  in  treatment. 

E.  H.  Barton,  Text-Book  of  Sound,  London  (1908),  687  pages.  An 
experimental  and  theoretical  treatise  on  sound  in  general. 

R.  Koenig,  Quelques  Experiences  d'Acoustique,  Paris  (1882),  248  pages. 
An  account  of  Koenig's  own  experimental  researches,  consisting  of  a 
collection  of  papers  published  in  various  scientific  journals  together  with 
others  not  published  elsewhere. 

A.  Winkelmann,  Handbuch  der  Physik,  2  aufl.,  Liepzig  (1909),  Bd.  II, 
Akustik,  F.  Auerbach,  714  pages.  An  encyclopedic  treatment  of  the 
whole  field  of  acoustics,  experimental  and  theoretical ;  contains  thousands 
of  references  arranged  according  to  subjects. 

J.  A.  Zahm,  Sound  and  Music,  Chicago  (1892),  452  pages.  A  popular, 
yet  scientific,  account  of  sound  in  general,  and  in  particular  with  reference 
to  music. 


271 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


Special  References  :  —  The  number  in  parenthesis,  following  a  sub- 
ject, refers  to  the  page  of  this  book  where  the  subject  is  treated. 

1.  Velocity  of  sound  (6).  Barton,  Text-Book  of  Sound,  pp.  513-553. 
Winkelmann,  Akustik,  S.  494-588,  a  full  discussion  of  velocity,  with  more 
than  200  references.  J.  VioUe,  Congres  International  de  physique,  I,  Paris 
(1900),  pp.  228-250. 

2.  Simple  harmonic  motion  (7).  J.  D.  Everett,  Vibratory  Motion  and 
Sound,  London  (1882),  pp.  73-83. 

3.  New  device  for  producing  simple  harmonic  motion  (11).  Mr.  J. 
C.  Smedley,  of  Cleveland,  in  1912,  devised  the  simple  harmonic  move- 
ment shown,  as  a  result  of  his  interest  in  the  harmonic  synthesizer  de- 
scribed in  Lecture  IV. 

4.  Sidney  Lanier  (24).  Letters  of  Sidney  Lanier,  New  York  (1899), 
p.  68. 

5.  The  siren  (28).    Helmholtz,  Sensations  of  Tone,  p.  161. 

6.  Determination  of  pitch  (28).  Winkelmann,  Akustik,  S.  178-227, 
with  many  references.    Barton,  Text-Book  of  Sound,  pp.  560-580. 

7.  Tuning  forks  (29).  Winkelmann,  Akustik,  S.  345-367,  an  extended 
account,  with  more  than  100  references.  E.  A.  Kielhauser,  Die  Stimm- 
gabel,  Leipzig,  (1907),  188  pages.  A.  J.  Ellis,  Appendix  to  Helmholtz' s 
Sensation  of  Tone,  pp.  443-446.  R.  Hartmann-Kempf,  Elektro-Akus- 
tische  Untersuchungen,  Frankfort  (1903),  255  pages,  with  many  plates. 

8.  Tuning-fork  resonator  (31).  R.  Koenig,  Quelques  Experiences, 
p.  180. 

9.  Temperature  coefficient  of  tuning  fork  (31).  R.  Koenig,  Annalen 
der  Physik,  9,  408  (1880) ;  Quelques  Experiences,  p.  182. 

10.  Relation  of  amplitude  and  pitch  of  tuning  fork  (32).  R.  Hart- 
mann-Kempf, Electro-Akustische  Untersuchungen,  Frankfort  (1903),  S. 
28-64. 

11.  Electrically  driven  tuning  fork  (33).  Rayleigh,  Theory  of  Sound,  I, 
pp.  65-69.    Barton,  Text-Book  of  Sound,  p.  361. 

272 


APPENDIX 


12.  Scheibler's  tonometer  (37).  Helmholtz,  Sensations  of  Tone,  pp. 
199,  443. 

13.  Koenig's  tonometer  (37).    Zahm,  Sound  and  Music,  p.  74. 

14.  Lissajous's  figures  (20,  37).  J.  Lissajous,  Comptes  Rendus,  Acad. 
Sci.  Paris  (1855).  Winkelmann,  Akustik,  S.  42-59.  These  figures  were 
described  by  Lissajous  in  1855,  but  they  had  been  previously  described 
by  Nathaniel  Bowditch,  of  Salem,  in  1815 ;  Mem.  American  Academy  of 
Arts  and  Sciences,  3,  413  (1815).  J.  Lovering,  Proc.  American  Academy 
of  Arts  and  Sciences,  N.  S.  8,  pp.  292-298. 

15.  French  pitch  (38).  R.  Koenig,  Quelques  Experiences,  p.  190; 
Annalen  der  Physik,  9,  394-417  (1880). 

16.  The  clock-fork  (38).  Niaudet,  Comptes  Rendus,  Acad.  Sci.  Paris, 
Dec.  10  (1866).    Koenig,  Quelques  Experiences,  p.  173. 

17.  Flicker  (44).  S.  H.  and  P.  H.  Gage,  Optic  Projection,  Ithaca 
(1914),  pp.  423-427. 

18.  Highest  audible  sound  (45).  R.  Koenig,  Annalen  der  Physik,  69, 
626-660,  721-738  (1899). 

19.  History  of  pitch  (49).  A.  J.  Ellis,  Appendix  Helmholtz' s  Sensa- 
tions of  Tone,  pp.  493-513. 

20.  Musical  pitch  in  America  (49).  Charles  R.  Cross,  Proc.  American 
Academy  of  Arts  and  Sciences,  35,  453-467  (1900). 

21.  Loudness  of  sound  (53).  Winkelmann,  Akustik,  S.  228-254.  A.  G. 
Webster,  Physical  Review,  16,  248  (1903). 

22.  Acoustic  properties  of  auditoriums  (57),  W.  C.  Sabine,  American 
Architect,  68  (a  series  of  papers)  (1900) ;  Proc.  American  Academy  of  Arts 
and  Sciences,  42,  51-84  (1906) ;  American  Architect,  104,  252-279  (1913). 

23.  Acoustic  properties  of  auditoriums  (58).  F.  R.  Watson,  Bulletin 
No.  78,  Engineering  Experiment  Station,  University  of  Illinois,  32  pages 
(1914);  gives  references  to  forty-one  papers  on  architectural  acoustics; 
Physical  Review,  6,  56  (1915). 

24.  Acoustic  properties  of  auditoriums  (58).  F.  P.  Whitman,  Science^ 
38,  707  (1913) ;  42,  191-193  (1915). 

T  273 


THE  SCIENCE  OF  MUSICAL  SOUNDS 

25.  Model  for  combinations  of  waves  (59).  E.  Grimsehl,  Zeitschrift 
f.  d.  physikalischen  u.  chemischen  Unterncht,  17,  34  (1904) ;  Physikalische 
Zeitschrift  (1904) ;  Frick's  Physikalische  Technik,  Leipzig  (1905),  I, 
S.  1346. 

26.  Tone  quality  (62).    Helmholtz,  Sensations  of  Tone,  p.  33. 

27.  Phase  and  tone  quality  (62).  Helmholtz,  Sensations  of  Tone, 
p.  126. 

28.  Phase  and  tone  quality  (63).  R.  Koenig,  Annalen  der  Physik,  57, 
555-566  (1896). 

29.  Phase  and  tone  quality  (63).  F.  Lindig,  Annalen  der  Physik,  10, 
242-269  (1903). 

30.  Phase  and  tone  quality  (63).  M.  G.  Lloyd  and  P.  G.  Agnew, 
Bulletin  of  the  Bureau  of  Standards,  6,  255-263  (1909). 

31.  Phase  and  tone  quality  (63).  Winkelmann,  Akustik,  S.  268-278. 
Barton,  Text-Book  of  Sound,  pp.  605-607. 

32.  Resonators  (68).  Helmholtz,  Sensations  of  Tone,  pp.  36-49,  372; 
Vorlesungen,  S.  246.    Rayleigh,  Theory  of  Sound,  II,  pp.  170-235. 

33.  The  phonoautograph  (71).    Leon  Scott,  Cosmos,  14,  314  (1859). 

34.  Manometric  flames  (73).  R.  Koenig,  Annalen  der  Physik,  146, 
161  (1872) ;  Quelques  Experiences,  pp.  47-70. 

35.  Photographing  manometric  flames  (74).  Nichols  and  Merritt, 
Physical  Review,  7,  93-101  (1908). 

36.  Vibrating  flames  (75).  J.  G.  Brown,  Physical  Review,  33,  442-446 
(1911). 

37.  The  oscillograph  (75).  A.  Blondel,  Comptes  Rendus,  Acad.  Sci. 
Paris,  116,  502,  748  (1893).  W.  Duddefl,  Proc.  British  Association  for  the 
Advancement  of  Science,  Toronto  (1897),  p.  575. 

38.  OsciUograph  records  (76).  D.  A.  Ramsey,  The  Electrician,  Sept. 
21  (1906). 

39.  The  phonograph  for  acoustical  research  (77).  L.  Hermann, 
Pfiuger's  Archiv,  45,  282  (1889) ;  47,  42,  44,  347  (1890) ;  and  others. 

274 


APPENDIX 


40.  The  phonograph  for  acoustical  research  (77).  L.  Bevier,  Physical 
Review,  10,  193  (1900). 

41.  Enlarging  phonograph  records  (77).  E.  W.  Scripture,  Experi- 
mental Phonetics,  Washington  (1906),  204  pages. 

42.  The  phonodeik  (79).  D.  C.  Miller,  Physical  Review,  28,  151 
(1909) ;  Science,  29,  471  (1909) ;  Proc.  British  Association  for  the  Advance- 
ment of  Science,  Winnipeg  (1909),  p.  414;  Proc.  British  Association  for 
the  Advancement  of  Science,  Dundee  (1912),  p.  419;  Engineering,  London, 
94,  550  (1912). 

43.  The  demonstration  phonodeik  (85).  D.  C.  Miller,  before  the 
American  Physical  Society  and  the  American  Association  for  the  Advance- 
ment of  Science,  Boston  meeting,  Dec.  1909. 

44.  Photographing  waves  of  compression  (88).  A.  Toepler,  Armalen 
der  Physik,  127,  556  (1866);  131,  33,  180  (1867).  E.  Mach,  Wiener 
Berichte,  77,  78,  92.  R.  W.  Wood,  Philosophical  Magazine,  48,  218 
(1899) ;  Physical  Optics,  2  ed..  New  York  (1911),  page  94.  Foley  and 
Souder,  Physical  Review,  35,  373-386  (1912).  W.  C.  Sabine,  American 
Architect,  104,  257-279  (1913). 

45.  Fourier's  Series  (92,  134).  J.  B.  J.  Fourier,  La  Theorie  Analytique 
de  la  Chaleur,  Paris  (1822) ;  The  Analytical  Theory  of  Heat,  English 
translation  by  Alexander  Freeman,  Cambridge  (1878).    466  pages. 

46.  Fourier's  Series  (93).  W.  E.  Byerly,  Fourier's  Series  and  Spheri- 
cal Harmonics,  Boston  (1893).  H.  S.  Carslaw,  Fourier's  Series  and 
Integrals,  London  (1906).  C.  P.  Steinmetz,  Engineering  Mathematics,  2 
ed..  New  York  (1915),  pp.  94-146.  Franklin,  McNutt,  and  Charles, 
Calculus,  South  Bethlehem  (1913),  pp.  199-209.  Carse  and  Shearer, 
Fourier's  Analysis  and  Periodogram  Analysis,  London  (1915). 

47.  Fourier's  Series  (97).  C.  "P.  Steinmetz,  Engineering  Mathematics, 
2  ed.,  New  York  (1915),  p.  112. 

48.  Henrici's  harmonic  analyzer  (98).  0.  Henrici,  Philosophical 
Magazine,  38,  110  (1894).  H.  de  Morin,  Les  Appariels  d' Integration, 
Paris  (1913),  pp.  162,  171.  E.  M.  Horsburgh,  Modern  Instruments  of 
Calculation,  London  (1914),  p.  223. 

275 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


49.  Instruments  for  harmonic  analysis  and  synthesis  (100,  104,  114). 
D.  C.  Miller,  Journal  of  the  Franklin  Institute  (1916). 

50.  Harmonic  analyzer  and  synthesizer  (129).  Lord  Kelvin,  Proc. 
Royal  Society,  27,  371  (1878) ;  Kelvin  and  Tait,  Natural  Philosophy, 
Part  I,  Appendix  B',  VII,  Cambridge,  (1896) ;  Kelvin,  Popular  Lectures, 
Vol.  Ill,  p.  184. 

51.  Tide-predicting  machine  (131).  E.  G.  Fisher,  Engineering  News, 
66,  69-73  (1911).  Special  Publication  No.  32,  United  States  Coast  and 
Geodetic  Survey,  Washington  (1915). 

52.  Harmonic  analyzer  and  synthesizer  (131).  Michelson  and  Strat- 
ton,  American  Journal  of  Science,  5,  1-13  (1898) ;  Philosophical  Maga- 
zine, 45,  85  (1898) ;  Michelson,  Light  Waves  and  Their  Uses,  Chicago 
(1903),  p.  68. 

53.  Harmonic  analyzer  (132).  G.  R.  Rowe,  Electrical  World,  March  25 
(1905).  0.  Mader,  Elektrochemische  Zeitschrift,  Nr.  36  (1909);  theory 
given  by  A.  Schreiber,  Physikalische  Zeitschrift,  11,  354  (1910).  L.  W. 
Chubb,  The  Electric  Journal  (Pittsburgh),  Feb.  1914,  May,  1914. 

54.  Various  harmonic  analyzers  (132,  134).  Carse  and  Urquhart,  Hors- 
burgh's  Modern  Instruments  of  Calculation,  London  (1914),  pp.  220- 
253,  337.  H.  de  Morin,  Les  Appariels  d' Integration,  Paris  (1913),  pp. 
147-188.  W.  Dyck,  Catalogue  (Munich  Mathematical  Exposition), 
Munich  (1892).  E.  Orlich,  Aufnahme  und  Analyse  von  Wechselstrom- 
kurven,  Braunschweig  (1906).  G.  U.  Yule,  Philosophical  Magazine,  39, 
367-374  (1895).  J.  X.  LeConte,  Physical  Review,  7,  27-34  (1898).  T. 
Terada,  Zeitschrift  fUr  Instrumentenkunde,  25,  285-289  (1905). 

55.  Harmonic  analysis,  in  meteorology  (133).  Strachey,  Proc.  Royal 
Society,  42,  61-79  (1887).  Steinmetz,  Engineering  Mathematics,  New 
York  (1915),  p.  125. 

56.  Harmonic  analysis  in  astronomy  (133).  A.  Schuster,  Terrestrial 
Magnetism,  3,  13  (1898).  H.  H.  Turner,  Tables  for  Facilitating  Harmonic 
Analysis,  Oxford  (1913).  A.  A.  Michelson,  Astrophysical  Journal,  38, 
268-274  (1913).  A.  E.  Douglass,  Astrophysical  Journal,  40,  326-331 
(1914).  Carse  and  Shearer,  Fourier's  Analysis  and  Periodogram  Analysis, 
London  (1915),  p.  34. 

276 


APPENDIX 


57.  Harmonic  analysis  in  mechanical  engineering  (133).  S.  P.  Thomp- 
son, Proc.  Physical  Society,  London,  33,  334-343  (1911).  W.  E.  Dalby, 
Valves  and  Valve-Gear  Mechanism,  London  (1906),  pp.  328-353. 

58.  Periodogram  analysis  for  non-periodic  curves  (133,  141).  Carse 
and  Shearer,  Fourier^s  Analysis  and  Periodogram  Analysis,  London 
(1915),  66  pages.    See  also  references  No.  56. 

59.  Harmonic  analysis  (134).  C.  P.  Steinmetz,  Engineering  Mathe- 
matics, 2  ed..  New  York  (1915),  pp.  114-134. 

60.  Harmonic  analysis  (134).  C.  Runge,  Zeitschrift  fiir  Mathematik 
und  Physik,  48,  443-456  (1903),  52,  117-123"  (1905) ;  Erlduterung  des 
Rechnungsformulars,  Braunschweig  (1913). 

61.  Harmonic  analysis  (134).  BedeU  and  Pierce,  Direct  and  Alter- 
nating Current  Manual,  2  ed..  New  York  (1911),  pp.  331-344. 

62.  Harmonic  analysis  (135).  F.  W.  Grover,  Bulletin  of  the  Bureau 
of  Standards,  9,  567-646  (1913).  H.  0.  Taylor,  Physical  Review,  6,  303- 
311  (1915). 

63.  Harmonic  analysis  (135).  S.  P.  Thompson,  Proc.  Physical  Society, 
London,  19,  443-450  (1905),  33,  334-343  (1911). 

64.  Graphical  method  for  harmonic  analysis  (135).  J.  Perry,  The 
Electrician,  35,  285  (1895).  A.  S.  LangsdoTf,  Physical  Review,  12,  184- 
190  (1901).  W.  R.  Kelsey,  Physical  Determinations,  London  (1907), 
pp.  86-93. 

65.  Graphical  methods  for  harmonic  analysis  (135).  Carse  and 
Urquhart,  Horsburgh's  Modern  Instruments  of  Calculation,  London 
(1914),  pp.  247,  248;  various  articles  in  The  Electrician  (1895),  (1905), 
(1911). 

66.  Resonance  effects  in  records  of  sounds  (143).  D.  C.  Miller,  Proc. 
Fifth  International  Congress  of  Mathematicians,  Cambridge  (1912),  II, 
pp.  245-249. 

67.  Vibrating  diaphragms  and  sand  figures  (151).  E.  F.  F.  Chladni, 
Theorie  des  Klanges,  Leipzig  (1787).    Winkelmann,  Akustik,  S.  368-401. 

68.  Resonance  (177).  E.  H.  Barton,  Text-Book  of  Sound,  pp.  146-148. 
Helmholtz,  Sensations  of  Tone,  pp.  143,  405. 

277 


THE  SCIENCE  OF  MUSICAL  SOUNDS 


69.  Effect  of  resonator  on  tuning  fork  (178).  R.  Koenig,  Quelques 
Experiences,  p.  180. 

70.  Damped  vibration  (179).  E.  H.  Barton,  Text-Book  of  Sound, 
pp.  91-96. 

71.  Material  and  tone  quality  (180).  C.  von  Schafhautl,  Allgemeine 
Musikalische  Zeitung,  Leipzig  (1879),  S.  593-599,  609-632. 

72.  Material  and  tone  quality  (180).  D.  C.  Miller,  Science,  29,  161- 
171  (1909). 

73.  Beat-tones  (183).    Zahm,  Sound  and  Music,  pp.  322-340. 

74.  Simple  tones  (185).    Helmholtz,  Sensations  of  Tone,  p.  70. 

75.  Octave  overtones  in  tuning  forks  (189).  E.  H.  Barton,  Text-Book 
of  Somid,  pp.  395-399. 

76.  The  Choralcelo  (189).    The  Music  Trade  Review,  April  29,  1911. 

77.  The  flute  (191).  Theobald  Boehm,  The  Flute  and  Flute-Playing  in 
Acoustical,  Technical,  and  Artistic  Aspects,  English  translation  by  D.  C. 
Miller,  Cleveland  (1908),  100  pages. 

78.  Sidney  Lanier  (191,  202).  Poems  of  Sidney  Lanier,  New  York, 
(1903),  page  62. 

79.  Vibration  of  violin  strings  (194).  Helmholtz,  Sensations  of  Tone, 
pp.  74r-88,  384-387 ;  Vorlesungen,  S.  121-139. 

80.  Vibration  of  violin  strings  (194).  H.  N.  Davis,  Proc.  American 
Academy  of  Arts  and  Sciences,  41,  639-727  (1906) ;  Physical  Review,  22, 
121  (1906),  24,  242  (1907).  ' 

81.  Vibration  of  violin  strings  (194).  E.  H.  Barton,  Text-Book  of 
Sound,  pp.  416-436 ;  and  numerous  papers  in  the  Philosophical  Magazine 
for  1906,  1907,  1910,  and  1912. 

82.  Violin  tone  quality  (195).  P.  H.  Edwards,  Physical  Review,  32, 
23-37  (1911).    C.  W.  Hewlett,  Physical  Review,  36,  359-372  (1912). 

83.  Theory  of  vowels  (216).  A  general  discussion  of  vowel  theories, 
together  with  references  to  the  published  works  of  a  very  long  list  of 
investigators,  will  be  found  in  the  following  books  :  Helmholtz,  Sensations 

278 


APPENDIX 


of  Tone,  pp.  103-126.  Lord  Rayleigh,  Theory  of  Sound,  vol.  II,  pp.  469- 
477.  Winkelmann,  Akustik,  S.  681-705,  contains  about  two  hundred 
references  to  papers  by  more  than  a  hundred  authors. 

84.  Analysis  of  vowels  (217).  L.  Bevier,  Physical  Review,  10,  193, 
(1900) ;  14,  171,  214  (1902) ;  15,  44,  271  (1902) ;  21,  80  (1905). 

85.  Theory  of  vowel  quality  (217).  E.  W.  Scripture,  Researches  in 
Experimental  Phonetics,  Washington  (1906),  pp.  7,  109. 

86.  Physical  characteristics  of  the  vowels  (217).  D.  C.  Miller,  before 
the  American  Physical  Society  and  the  American  Association  for  the 
Advancement  of  Science,  Atlanta  meeting  (1913-1914). 

87.  Vowel  characteristics  (240).  R.  Koenig,  Comptes  Rendus,  Acad. 
Sci.  Paris,  70,  931  (1870,  Quelques  Experiences,  p.  42). 

88.  Artificial  vowels  (244).  Marage,  Physiologie  de  la  Voix,  Paris, 
(1911),  p.  92. 

89.  The  wave-siren  (245).  R.  Koenig,  Annalen  der  Physik,  57,  339- 
388  (1896). 

90.  Tuning-fork  synthesis  of  tones  (245).  Helmholtz,  Sensations  of 
Tones,  pp.  123-128,  398-400. 


279 


INDEX 


A 


Acoustics  of  auditoriums,  56,  89,  264. 
Agnew  and  Lloyd,  phase  and  tone  quality, 
63. 

Alternating  current,  for  phase  effect,  63  ; 

for  tuning  fork,  32  ;  waves,  137. 
Amplitude,  7,  8,  53  ;  effect  of,  on  pitch 

of  tuning  fork,  32  ;  related  to  loudness, 

53,  144. 

Amplitude  and  phase  calculator,  103,  123. 

Analysis,  arithmetical  and  graphical,  133  ; 
harmonic,  92  ;  by  inspection,  136  ;  of 
organ-pipe  curve,  125  ;  of  violin  curve, 
101,  103;  wave  method  of,  92. 

Analyzer,  harmonic,  97 ;  Henrici,  98, 
108;  various,  128. 

Arithmetical  harmonic  analysis,  133. 

Art  and  science  of  music,  1,  262. 

Art  of  piano  playing,  208,  264. 

Artificial  vowels,  244, 

Atmospheric  vibration,  2,  17,  20. 

Audibility,  limits  of,  42. 

Auditoriums,  acoustics  of,  56,  89,  264. 

Auerbach,  theory  of  vowels,  216. 

Automatic  piano,  208,  264. 

Axis  of  curve  determined,  107,  124 ; 
photographed,  82. 


Bach,  equally  tempered  scale,  64,  264, 
265. 

Barton,  violin  string,  194. 

Bass  voice,  205  ;  vowels,  229. 

Beats,  33,  87,  138,  183. 

Beat-tone,  62,  63,  183  ;  of  violin,  198. 

Bedell  and  Pierce,  harmonic  analysis,  134. 

Bell,  62  ;  glass,  4  ;  photograph  of  sound 

of,  141. 
Bell,  telephone,  75. 
Bell  and  Tainter,  graphophone,  77. 
Berliner,  gramophone,  77. 
Berlioz,  the  violin,  196. 
Bevier,  analysis  of  phonograph  records, 

77;  theory  of  vowels,  217. 


Bibliography,  Appendix,  271. 
Blondell,  oscillograph,  75. 
Boehm,  inventor  of  flute,  191,  263. 
Boston  Symphony  Orchestra,  adopts  inter- 
national pitch,  50. 
Bottles,  tuned,  22. 
Bowing,  violin,  reversal  of,  197. 
Brown,  manometric  flame,  75. 
Bugle,  67. 

C 

Camera  for  photographing  sound  waves, 
82. 

Capsule,  manometric,  73. 

Cards  for  records  of  harmonic  analysis, 

122,  123,  165,  167. 
Carse  and  Urquhart,  harmonic  analysis, 

134. 

Centennial  Exposition,  Philadelphia, 
Koenig's  exhibit,  46 ;  Grand  March 
for,  267. 

Century  Dictionary,  list  of  vowels,  217. 

Characteristic  noises,  24,  185. 

Characteristics  of  vowels,  215,  225,  259  ; 
see  vowels. 

Chart  for  sound  analysis,  168. 

Chladni,  sand  figures,  151,  153. 

Choralcelo,  189. 

Chromatic  scale,  35,  48. 

Clarinet,  176 ;  curve,  analyzed  by  in- 
spection, 138  ;  tone  quality  of,  199,  251. 

Classification  of  vowels,  225  ;  see  vowels. 

Clifford,  harmonic  analysis,  134. 

Clock-fork,  28,  38,  50. 

Cobb,  L.  N.,  phonodeik,  82. 

Coefficient,  in  Fourier  equation,  94,  97, 
123;  temperature,  of  fork,  31. 

Collision  balls,  18. 

Color,  tone,  25,  58 ;  see  tone  quality. 
Compound  pendulum,  18. 
Compression  wave,  17  ;  photographs  of, 
88. 

Convergence  of  series,  115. 

Correction  of  analyses,  162  ;  curve,  163  ; 

of  sound  waves,  172. 
Cosine  curve,  11. 


281 


INDEX 


Costa,  Sir  Michael,  philharmonic  pitch,  49. 

Cross,  history  of  pitch,  50. 

Curve,  axis  of  determined,  107,  124 ; 
correction,  163 ;  cosine,  11  ;  energy, 
170 ;  enlarging,  108 ;  graphical  study 
of,  115;  simple  harmonic,  11;  sine, 
11  ;   synthesis  of  harmonic,  110. 

D 

Damped  vibration,  179. 

Davis,  violin  string,  194. 

Demonstration  phonodeik,  85,  214. 

Diagram  of  sound  analysis,  169. 

Diapason  Normal,  50. 

Diapason  organ  pipes,  145. 

Diaphragm,  70  ;  effect  of  on  sound  rec- 
ords, 142  ;  response  of,  148  ;  influence 
of  diameter,  149  ;  effect  of  clamping, 
150 ;  modes  of  vibration,  151  ;  free 
periods  of,  153  ;  influence  of  mount- 
ing, 155. 

Dickinson,  "Music  and  Higher  Educa- 
tion," 266. 

Displacement,  longitudinal  and  trans- 
verse, 16. 

Donders,  theory  of  vowels,  215. 

Drum,  71. 

Duddell,  oscillograph,  75. 

E 

Ear,  20,  22,  53,  62,  68,  70;  pitch  de- 
termined by,  33. 
Edison,  phonograph,  76. 
Edwards  and  Hewlett,  violin  string,  194. 
Elasticity,  6,  14. 

Electro-magnetic  operation  of  tuning 
fork,  32. 

Ellis,  tonometer,  37  ;  history  of  pitch, 
49  ;  theory  of  vowels,  216. 

Energy  of  sound,  53,  100,  167,  179 ;  dis- 
tribution of,  170,  220;  in  piano  tone, 
179  ;  of  vowels,  226,  227. 

Engineering,  harmonic  analysis  in,  133, 
135. 

Enlarging  curves,  108. 
Epoch  of  component  of  curve,  126 ;  see 
phase. 

Equally  tempered  scale,  table  of  fre- 
quencies, 48 ;  compared  with  har- 
monics, 64 ;  invention  of,  264,  265 ; 
tuning  forks  for,  34. 

Equation,  Fourier's,  93,  140. 

Errors  in  sovmd  records,  142  ;  correcting, 
162. 


Everett,    device    for    simple  harmonic 

motion,  10. 
Explosion  of  skyrocket,  photographed, 

139. 

Explosive  sounds,  6,  139,  217. 

F 

Figures,  Lissajous's,  20,  28,  37,  38,  41, 
249. 

Fine  arts,  1,  265,  268. 

Fireworks,  photographed,  139. 

Fixed  pitch  theory,  of  vowels,  216,  257 ; 

of  instruments,  257. 
Flame,  manometric,  73. 
Flicker,  in  moving  pictures,  43. 
Flute,  2,  23,  68,  176 ;  analysis  of  tone, 

171 ;   effect  of  material  of,  180,  192 ; 

of  gold,  192 ;    simple  tone  of,  185 ; 

tone  compared  to  tuning  fork,  185,  to 

voice,  258 ;  tone  quality  of,  190,  251. 
Foley  and  Souder,  compression  waves,  88. 
Forced  vibration,  177. 
Fork,  tuning,  see  tuning  fork. 
Fourier,  theorem,  92,  115.  122,  140. 
Franklin  Institute,  Journal  of,  114. 
Free  period,  71,  143,  153,  158,  177. 
French  horn,  tone  quality  of,  202,  213. 
French  vowels,  218. 

Frequency,  7,  25,  26  ;  and  loudness,  53, 

144  ;  see  pitch. 
Fuller,  Gen.  Levi  K.,  musical  pitch,  50. 
Fundamental  tones,  62. 
Funeral  March,  Siegfried,  267. 

G 

Gallon,  whistle,  44. 

Generator,  sound,  175. 

Geophysics,  harmonic  analysis  in,  133. 

German  vowels,  218. 

Glass  bell,  4. 

Gramophone,  77,  see  talking  machine. 
Graphical  harmonic  analysis,  133,  135. 
Graphical  presentation  of  analyses,  166, 
219. 

Graphophone,  77,  see  talking  machine. 
Grassmann,  theory  of  Vowels,  215. 
Grover,  harmonic  analysis,  134. 

H 

Hallelujah  Chorus,  Messiah,  262. 
Handel,  pitch,  49  ;  trumpeter,  29. 
Harmonic  analysis,  92  ;  arithmetical  and 
graphical,  133  ;  complete  process,  120  ; 


INDEX 


example  of,  122  ;  by  inspection,  136  ; 
limitations,  140 ;  by  machine,  97 ; 
verified  by  synthesis,  12S. 
Harmonic  analj-zer,  Kelvin's,  12N; 
Michelson's,  131:  Rowe's,  132; 
Mader's,  132;  Chubb's,  132;  Hen- 
rici's,  98  ;  extended  to  30  components, 
100. 

Harmonic  curves,  11.  92. 

Harmonic  plotting  scale,  165,  169,  228. 

Harmonics,  62  ;  tune  in,  68. 

Harmonic  synthesizers,  110;  Michel- 
son's,  131. 

Harmony,  study  of,  265. 

Hauptmann,  musical  critic,  23. 

HelmJxoltz,  art  of  music,  265  ;  law  of  tone 
quality,  62,  63  ;  limits  of  audibility, 
42,  43 ;  resonance  of  mouth,  240 ; 
resonators,  68 ;  simple  tones,  185 ; 
theory  of  vowels,  215  ;  violin  string, 
194  :  vowel  apparatus,  245,  263. 

Henrici,  harmonic  analyzer,  98,  120. 

Hermann,  analysis  of  phonograph  records, 
77  ;  theory  of  vowels.  216. 

Hewlett  and  Edwards,  violin  string,  194. 

Hiller,  Ferdinand,  composer,  269. 

Horn,  French,  tone  quality  of,  202,  213. 

Horn,  resonating,  effect  of,  on  sound 
records,  142,  156  ;  flare  of,  160  ;  length 
of,  159 ;  of  various  materials,  157 ; 
resonance  of.  161. 

Horsburgh,  "Instruments  of  Calcula- 
tion," 132,  135. 

Hughes,  microi)hone,  75. 

I 

Ideal  musical  tone.  204.  212. 
Inharmonic  components,  113,  141. 
Inharmonic  partials,  62,  141,  188,  201. 
Inspection,  harmonic  analysis  by,  136. 
Integrator,  97,  98,  108. 
Intensity  of  sound,  25,  53 ;    of  simple 

sound.  144  ;  see  energy. 
International  pitch,  49,  50. 
Interrupter  fork,  32. 

K 

Kelvin,  Lord,  20 ;  harmonic  analyzer, 
128;  harmonic  synthesizer,  lib ;  tide 
predictor,  129. 

Kintner,  harmonic  analysis,  134. 

Koenig,  clock-fork,  38,  50;  limits  of 
audibility,  45,  46  ;  manometric  flame, 
73 ;     phase    and    tone    qualitj',    62 ; 


phonautograph,  71  ;  resonance  box 
for  fork,  177 ;  resonance  of  mouth, 
240;  scientific  pitch.  51;  tonometer, 
37;  tuning  forks,  29,  31.  50;  vowel 
characteristics.  240  ;  wave  sircMi.  244. 

L 

Lanier,   quotations  from   writings,  24, 

191,  202,  268,  269. 
Lavignac,  the  horn,  202. 
Leyden  jar,  photographing  sound  from, 

88. 

Limits  of  pitch.  42. 

Lindig,  phase  and  tone  quality,  63. 

Lissajous.  figures,  20,  28,  37.  38,  41; 

method  of  tuning,  249. 
Lloyd  and  Agnew,  phase  and  tone  qualitv, 

63. 

Lloyd,  theory  of  vowels.  216. 

Logarithmic  scale,  145.  147.  168. 

Longitudinal  displacement.  16.  20 :  vi- 
bration, 3  ;  wave.  15. 

Loudness,  25,  53  ;  of  simple  sound.  144  ; 
see  energy. 

Lowell  Institute,  photograph  of  the  words, 
239,  255. 

Lucia  di  Lammermoor,  Mad  Scene,  194; 
Sextette,  frontispiece,  211,  239. 

M 

Mack,  compression  waves.  88. 

Mad  Scene  from  Lucia.  194. 

Manometric  capsule.  73. 

Manometric  flame,  73. 

Marage,  imitates  vowels.  244. 

Marloye,  inventor  of  resonance  box.  31. 

Material  affecting  sound  waves.  179. 

Maurier,  du,  "Peter  Ibbetsen,"  268. 

Mechanical,  harmonic  analysis,  97  ;  syn- 
thesis. 110;  calculation,  103,  168. 

Merritt  and  Xichols,  manometric  flames, 
74. 

Messiah.  "Hallelujah  Chorus,"  262. 
Meteorology,  harmonic  analysis  in.  133. 
Michelson,  harmonic  analyzer  and  syn- 
thesizer, 131. 
Microscope,  vibration,  38.  41,  195. 
'  Middle  C,  49. 
'  Molecular  vibration.  3. 
Morin,  "Les  Appariels  d'Integration," 
132. 

Motion,  of  one  dimension,  20,  85  :  pen- 
dular,  6  ;  simple  harmonic,  6  :  vibra- 
torj',  6  ;  wave,  13. 


INDEX 


Mouth,  resonance  of.  228.  239. 
Moving  picture  apparatus,  flicker,  43. 
Mozart,  i)itch  in  time  of,  49. 
Musical  scale,  47. 

Music,  art  of,  1  ;  science  of,  1 ;  science 
and  art  of,  262,  269  ;  photograph  of 
Sextette,  frontispiece. 


N 


Naval  architecture,   harmonic  analj^sis 
in,  133. 

Xewton,  "Mysticism  of  Music,"  267. 
Xiaiidet,  clock-fork,  38. 
Nichols  and  Merritt,  manometric  flames, 
74. 

Nodes,  4  ;    in  a  string,  67 ;    shown  by 

sand  figures,  151. 
Noise,  21,  24  :  characteristic,  24,  185. 
Non-periodic  and  periodic  curves,  140 ; 

vibrations,  22. 


Oboe,  tone  quality  of,  199,  251. 
Ohm,  law  of  acoustics,  62,  140. 
Opera  in  English,  259. 
Optical  method,  Lissajous's,  20,  28,  37, 
41,  249. 

Order,  of  partials,  63  ;   of  components, 

97,  137. 
Organ,  33,  42. 

Organ  pipe,  analysis  and  synthesis  of 
sound  wave,  122,  127;  largest,  42; 
open  diapason  and  stopped  diapason, 
145 ;  smallest,  44 ;  tibia,  247 ;  for 
sound  synthesis,  246  ;  of  uniform  loud- 
ness, 146  ;  of  various  materials,  180. 

Oscillograph.  75. 

Overtones,  02,  64  :  combination  of,  212. 


Pantograph,  10. 

Paris,  Conservatory  of  Music  and  Grand 

Opera,  38. 
Partials,  inharmonic,  62,  141,  201. 
Partial  tones,  62,  140,  168,  175. 
Pendular  motion,  6,  12. 
Pendulum,  simple,  12  ;  compound,  19. 
Period,  7,  8 ;  see  free  period,  frequency, 

pitch. 

Periodic  and  nonperiodic  curves,  140; 

vibrations,  22. 
Perry,  harmonic  analysis,  134. 
Persistence  of  vision,  43. 


Phase,  defined,  7,  8  ;  does  not  affect  tone 
quahty,  166.  197  ;  effect  on  tone,  62  ; 
explained,  61,  126;  relative,  deter- 
mined with  synthesizer,  114. 

Phase  and  amplitude  calculator,  103. 

Phonautograph,  71. 

Phonodeik,  28 ;  described,  78 ;  charac- 
teristics of,  143 ;  for  demonstration. 
85 ;  for  determining  pitch,  87 ;  for 
tone  synthesis,  249  ;  for  tuning,  249. 

Phonograph,  76  ;  translation  of  vowels, 
232  ;  see  talking  machine. 

Piano,  33.  42,  70,  176,  178 ;  automatic, 
208,  264  ;  duration  of  sound,  179  ;  tone 
quahty  of,  207  ;  touch,  208. 

Pianola,  208,  264. 

Pierce  and  Bedell,  harmonic  analysis,  134. 
Pin-and-slot  device,  7,  110. 
Pitch,  25,  26 ;   American,  49 ;  concert, 
49  ;  determined  by  beats,  33  ;  deter- 
mined with  the  phonodeik,  87  ;  diapa- 
son normal,  50 ;    French,  50 ;  high, 
49  ;   international,  49  ;   Koenig's,  51  ; 
limits  of,  42  ;  low,  50 ;  philharmonic, 
49  ;  philosophical,  51 ;  scientific,  51 ; 
Stuttgart,  50  ;  see  frequency. 
Planimeter,  97,  107,  124. 
Player  piano,  208,  264. 
'  Plotting  sound  analyses,  166,  219. 
Portrait,   harmonic  analysis  and  syn- 
thesis of,  118,  141 ;  wave  form,  120. 
I  Prediction  of  tides,  129. 
I  Pressure  wave,  17,  70,  88. 
I  Profile,  harmonic  analysis  of,  119,  141. 
I  Pyramid,  classification  of  vowels,  230. 

Q 

Quahty  of  sound,  25,  58 ;  see  tone 
quality. 

R 

Rayleigh,  Lord,  photograph  of  the  words, 
239,  263  ;  theory  of  vowels,  216. 

Reed  instruments,  199. 

Reference  books,  Hst  of,  Appendix,  271. 

Relative  pitch  theorj-,  of  vowels,  216, 
257  ;  of  instruments,  257. 

Resonance,  176 ;  box  for  tuning  forks, 
31,  177;  curves,  148,  150;  of  horn, 
158 ;  sharpness  of,  177  ;  of  \'iohn  body, 
197 ;  of  vocal  ca\ities,  228,  239. 

Resonators,  69,  175,  178. 

Response  to  sound,  ideal.  144  ;  actual, 
145. 

Reverberation.  57.  58,  91. 


284 


INDEX 


Reversal  of  violin  bow,  197. 
Rosa,  harmonic  analysis,  134. 
Range,  harmonic  analysis.  134. 
Rust,  effect  of,  on  tuning  fork. 


S 


33. 


Sabine,  acoustics  of  auditoriums,  57,  88. 

Sand  figures,  Chladni,  151. 

Scale,  chromatic,  35  ;  equally  tempered, 

frequencies.  48  :  musical.  47. 
Scale,  harmonic   graduated,    165.  169. 

228  ;  logarithmic,  145,  147.  168. 
Sehafhantl.  influence  of  material  on  tone. 

180. 

Scheibler.  tonometer.  37,  50. 

Science  and  art  of  music,  1,  262.  269. 

Scott,  phonautograph.  71. 

Scripture,  analysis  of  phonograph  record. 

77:  theory  of  vowels.  217. 
Sensation.  L  2.  43.  53.  270. 
Senses.  1. 

Series,  Fourier,  93 ;  studied  with  syn- 
thesizer. 115. 

Sextette  from  Lucia,  frontispiece.  211.  239. 

Shore,  inventor  of  tuning  fork.  29. 

Siegfried,  Death  Music.  267. 

Silences,  of  musical  value.  268. 

Simple  harmonic  curve.  11. 

Simple  harmonic  motion,  defined,  6 : 
by  mechanical  movement.  7.  110. 

Simplified  spelling.  256. 

Sine  curv  es,  11,  137:  and  cosine  curves, 
compounded,  94.  102.  105  :  from  fork. 
187. 

Singing  tone.  179,  242,  259. 
Siren.  27.  28  :  wave,  62.  244. 
SkjTocket,  photographed,  139. 
Smart,  Sir  George,  philharmonic  pitch, 
49. 

Smedley,  device  for  simple  harmonic 
motion.  11. 

Soprano  voice,  frontispiece,  206,  211. 
239;  vowels,  228. 

Souder  and  Foley,  compression  waves,  88. 

Sound  analyses,  diagram  of,  169  ;  graph- 
ical presentation,  166. 

Soundboard  of  piano,  70,  176,  178,  207. 

Soimd,  defined,  2  ;  explosive,  6  ;  records, 
errors  in,  142  ;  velocity  of,  5  ;  waves, 
5 ;  waves  made  visible,  85,  88,  214. 

Sounding  body,  2. 

Spectrum,    compared    to  analysis 

sound,  172. 
Staff,  47. 

Standard  pitches,  49. 


of 


Statistics,  studied  by  harmonic  analysis, 
133. 

Steinmetz.  harmonic  analysis,  134. 

Sticks  of  wood,  tuned,  22. 

Strings,  vibrating,  64. 

Stuttgart,  standard  pitch,  50. 

Sympathetic  vibration.  178. 

Synthetic  vowels.  244.  250. 

Synthesis,  of  harmonic  analysis.    12.s ; 

of  harmonic  curves.   110;    of  tones. 

244.  251. 


Tainter  and  Bell,  graphophone.  77. 
Talking  macliine.  20.  70.  264  :  records 

analyzed.  77:  see  phonograph. 
Tannhauser  Overture.  23,  24. 
Taylor,  harmonic  analysis.  135. 
Telephone.  20.  70.  75.  264 :   for  phase 

experiments.  63  :  transmitting  vowels, 

232. 

Telephone  siren.  63. 

Temperature,  effect  on  tuning  fork.  31  : 

and  velocity,  5. 
Tempered  scale,   table  of  frequeni-ies. 

48 :    harmonics  compared  \s-ith.  (Vi ; 

invention  of.  264 :    tuning  forks  for. 

34. 

Theorem,  Fourier's.  92. 

Theorj'  of  vowels,  215,  239. 

Thomas,  Orchestra,  24. 

Thompson,  S.  P.,  harmonic  analysis.  134. 

Tibia  organ  pipes,  247. 

Tidal  analysis,  129,  135. 

Tide  predictor,  Keh-in's,  129 ;  of  United 

States  Coast  and  Geodetic  Survev, 

129. 

Time  signals  on  record  of  sound  waves, 

82,  87,  139. 
Time  required  for  harmonic  analysis, 

120,  136. 
Toepler,  compression  waves,  88. 
Tone,  25,  26 ;  and  noise,  21 ;  ideal,  204. 

212  ;  pure  tone  is  poor,  213. 


25,  58,  267,  269;   see  ton 


Tone  color, 
quality. 

Tone  quality,  25,  58,  70,  174,  175  ;  law 
of,  62 ;  independent  of  phase,  166, 
197 ;  synthetic,  251 ;  of  vowels,  59, 
215,  239. 

Tonometer,  Koenig's  and  Scheibler's, 
37. 

Torsional  wave,  20. 
Touch,  piano,  208. 

Translation,  of  Grand  Opera.  261  :  of 
vowels  with  phonograph.  232. 
285 


INDEX 


Transverse,  displacement,  16,  20  ;  vibra- 
tion, 4  ;  wave  motion,  14. 

Tuning  fork,  adjusting,  30,  34  ;  analysis 
of  sound  of,  171  ;  clock-fork,  28,  38  ; 
exciting,  32  ;  for  higher  limit  of  audi- 
bility, 46  ;  invention  of,  29  ;  for  lower 
limit  of  audibility,  43  ;  as  musical  in- 
strument, 189;  overtones  of .  138,  187, 
188 ;  pitch  affected  by  amplitude, 
32;  quality  of  tone,  137,  185,  186; 
resonance  box,  31,  177;  rust  and 
wear,  33  ;  shapes  of,  30  ;  synthesizer 
for  vowels,  245  ;  effect  of  temperature 
on,  31  ;  for  tuning  chromatic  scale, 
35  ;  vibrations  of,  3. 

Tuning,  instruments.  34,  51,  87,  177, 
185. 

U 

Urquhart  and  Carse,  harmonic  analvsis, 
134. 

V 

Velocity  of  sound,  5. 

Vibrating  strings,  64. 

Vibration,  atmospheric,  2  ;  damped,  179  ; 
forced,  177;  free,  177;  longitudinal, 
3;  microscope,  38,  41,  195;  sympa- 
thetic, 178  ;  transverse,  4. 

Violin,  33,  58,  68 ;  analysis  of  tone,  100, 
171 ;  resonance  of  body,  197  ;  reversal 
of  bow,  197 ;  tone  quality  of,  194, 
251. 

Vision,  persistence  of,  43. 

Voice,  analysis  of,  171  ;  comparison  of 
bass  and  soprano,  205  ;  compared  with 
instruments,  257 ;  quality,  58 ;  re- 
lated to  vowels,  228,  259. 


Vowel  curve,  analyzed  by  inspection, 
137  ;  periodic  curve,  140. 

Vowels,  analyses  of,  221 ;  artificial,  244  ; 
characteristics  of,  215,  225,  230 ;  clas- 
sification of,  225,  230 ;  continuity  of, 
230;  defined,  217;  diagram  of  analy- 
sis, 219  ;  list  of,_218^  257  ;  musical 
quality  of,  59 ;  oo,  ee,  231  ;  photo- 
graphing, 219  ;  relation  to  pitch,  221, 
258,  260  ;  singing  voice  related  to,  259  ; 
synthetic,  244;  theory  of,  215,  239; 
translation  with  phonograph,  232 ; 
from  different  voices,  224  ;  whispered, 
235  ;  words  formed  from,  24,  251,  257. 

W 

Wagner,  composer,  23,  24,  267. 

Watson,  acoustics  of  auditoriums,  58. 

Waves,  of  compression,  17  ;  longitudinal, 
15  ;  photographs  of  compression,  88  ; 
in  solids,  liquids,  and  gases,  17 ;  of 
sound,  5 ;  torsional,  20 ;  transverse, 
14;  portrait,  120. 

Wave  models,  14,  15,  18,  19,  59,  60,  61. 

Wave  motion,  13. 

Wave  siren,  62,  244. 

Wedmore,  harmonic  analysis,  134. 

Wheatstone,  theory  of  vowels,  215. 

Whispered  vowels,  235. 

Whitman,  acoustics  of  auditoriums,  58. 

Willis,  theory  of  vowels,  215. 

Wires  and  cords  in  an  auditorium,  58. 

Wood,  compression  waves,  88. 

Words,  formation  of,  24,  251,  257  ;  syn- 
thetic, 253,  255  ;  photographs  of,  238, 
254. 

Y 

Yodeling,  nature  of,  260. 


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